{"hits":{"hits":[{"metadata":{"documents":[{"filename":"document.pdf","attachment":{"content":"HAL Id: tel-00610015\nhttps://tel.archives-ouvertes.fr/tel-00610015v2\n\nSubmitted on 20 Dec 2011\n\nHAL is a multi-disciplinary open access\narchive for the deposit and dissemination of sci-\nentific research documents, whether they are pub-\nlished or not. The documents may come from\nteaching and research institutions in France or\nabroad, or from public or private research centers.\n\nL’archive ouverte pluridisciplinaire HAL, est\ndestinée au dépôt et à la diffusion de documents\nscientifiques de niveau recherche, publiés ou non,\némanant des établissements d’enseignement et de\nrecherche français ou étrangers, des laboratoires\npublics ou privés.\n\nStudy of planar pixel sensors hardened to radiations for\nthe upgrade of the ATLAS vertex detector\n\nMathieu Benoit\n\nTo cite this version:\nMathieu Benoit. Study of planar pixel sensors hardened to radiations for the upgrade of the ATLAS\nvertex detector. Other [cond-mat.other]. Université Paris Sud - Paris XI, 2011. English. �NNT :\n2011PA112070�. �tel-00610015v2�\n\nhttps://tel.archives-ouvertes.fr/tel-00610015v2\nhttps://hal.archives-ouvertes.fr\n\n\nLAL $$%$$& \nMai 2011 \n\n \n \n \n \n \n \n\nTHÈSE \n \n \n\nPrésentée le 10 juin 2011 \n \n\npar \n \n\nMathieu BENOIT \n \n \n\npour obtenir le grade de \n \n\nDocteur ès Sciences \nde l’Université Paris XI, Orsay \n\n \n \n \n \n\nÉtude des détecteurs planaires pixels durcis aux \nradiations pour la mise à jour du détecteur \n\nde vertex d’ATLAS \n \n \n\n \n \n \n \n \n \n\nSoutenue devant la commission d’examen composée de : \n \n\nM. E. Augé  \nM. G Casse Rapporteur \nM. C. Goessling Rapporteur \nM. L.-A. Hamel  \nM. A. Lounis Directeur de thèse \nM. A. Stocchi Président \n\n\n\n\"I’ve found from past experiences that the tighter your plan, the more likely you are to\nrun into something unpredictable.\"\n\nMacGyver\n\n\n\n\n\n4\n\n\n\nRésumé\n\nLe Large Hadron Collider (LHC), située au CERN, Genève, produit des collisions de\n\nprotons accélérés à une énergie de 3.5 TeV depuis le 23 Novembre 2009. L’expérience\n\nATLAS enregistre depuis des données et poursuit sa recherche de nouvelle physique à\n\ntravers l’analyse de la cinématique des événements issues des collisions. L’augmentation\n\nprévue de la luminosité sur la période s’étalant de 2011 2020 apportera de nouveaux\n\ndéfis pour le détecteur qui doivent être considérés pour maintenir les bonnes performance\n\nde la configuration actuelle. Le détecteur interne sera le sous-détecteur le plus affecté\n\npar l’augmentation de la luminosité qui se traduira par une augmentation des dommages\n\noccasionés par la forte radiation et par la multiplication du nombre de traces associées à\n\nchaque croisement de faisceau. Les dommages causés par l’irradiation intense entrainera\n\nune perte d’efficacité de détection et une réduction du nombre de canaux actifs.\n\nUn intense effort de Recherche et Developpement (R&D) est présentement en cours\n\npour concevoir un nouveau détecteur pixel plus tolérant aux radiations et au cumul des\n\névénements générant un grand nombre de traces à reconstruire. Un premier projet de mise-\n\nà-jour du détecteur interne, nommé Insertable B-Layer (IBL) consiste à ajouter un couche\n\nde détection entre le tube à vide du faisceau et la première couche de silicium. Le projet\n\nSLHC prévoit de remplacer l’ensemble du détecteur interne par une version améliorée plus\n\ntolérante aux radiations et aux cumuls des événements. Dans cet ouvrage, je présente\n\nune étude utilisant la simulation technologique assisté par ordinateur (TCAD) portant\n\nsur les méthodes de conception des détecteurs pixels planaires permettant de réduire les\n\nzones inactives des détecteurs et d’augmenter leurs tolérances aux radiations. Les différents\n\nmodèles physiques disponible ont étés étudiés pour développer un modèle cohérent capable\n\nde prédire le fonctionnement des détecteurs pixels planaires après irradiation. La structure\n\n5\n\n\n\nd’anneaux de gardes utilisée dans le détecteur interne actuel a été étudié pour obtenir de\n\nl’information sur les possible méthodes permettant de réduire l’étendu de la surface occupée\n\npar cette structure tout en conservant un fonctionnement stable tout au long de la vie du\n\ndétecteur dans l’expérience ATLAS. Une campagne de mesures sur des structures pixels fut\n\norganisée pour comparer les résultats obtenue grâce à la simulation avec le comportement\n\ndes structures réelles. Les paramètres de fabrication ainsi que le comportement électrique\n\nont été mesurés et comparés aux simulations pour valider et calibrer le modèle de simulation\n\nTCAD. Un modèle a été développé pour expliquer la collection de charge excessive observée\n\ndans les détecteurs planaires en silicium lors de leur exposition a une dose extrême de\n\nradiations.\n\nFinalement, un modèle simple de digitalisation à utiliser pour la simulation de per-\n\nformances detecteurs pixels individuels exposès à des faisceau de hauteènergie ou bien de\n\nl’ensemble du détecteur interne est présenté. Ce modèle simple permets la comparaison\n\nentre les données obtenue en faisceau test aux modèle de transport de charge inclut dans la\n\ndigitalisation. Le dommage dû à la radiation , l’amincissement et l’utilisation de structures\n\nà bords minces sont autant de structures dont les effets sur la collecte de charges affectent\n\nles performance du détecteur. Le modèle de digititalisation fut validé pour un détecteur\n\nnon-irradié en comparant les résultats obtenues avec les données acquises en test faisceau\n\nde hauténergie. Le modèle validé sera utilisé pour produire la première simulation de l’IBL\n\nincluant les effets d’amincissement du substrat, de dommages dûes aux radiations et de\n\nstructure dotés de bords fins.\n\nKeywords : Dommage induit par la radiation, silicium, détecteur pixel planaires, simu-\n\nlation TCAD, test faisceau , IBL, SLHC\n\n6\n\n\n\n\n\n8\n\n\n\nAbstract\n\nIn this work, is presented a study, using TCAD simulation, of the possible methods of\n\ndesigning of a planar pixel sensors by reducing their inactive area and improving their\n\nradiation hardness for use in the Insertable B-Layer (IBL) project and for SLHC upgrade\n\nphase for the ATLAS experiment. Different physical models available have been studied\n\nto develop a coherent model of radiation damage in silicon that can be used to predict\n\nsilicon pixel sensor behavior after exposure to radiation. The Multi-Guard Ring Structure,\n\na protection structure used in pixel sensor design was studied to obtain guidelines for the\n\nreduction of inactive edges detrimental to detector operation while keeping a good sensor\n\nbehavior through its lifetime in the ATLAS detector. A campaign of measurement of the\n\nsensor’s process parameters and electrical behavior to validate and calibrate the TCAD\n\nsimulation models and results are also presented. A model for diode charge collection in\n\nhighly irradiated environment was developed to explain the high charge collection observed\n\nin highly irradiated devices.\n\nA simple planar pixel sensor digitization model to be used in test beam and full detector\n\nsystem is detailed. It allows for easy comparison between experimental data and prediction\n\nby the various radiation damage models available. The digitizer has been validated using\n\ntest beam data for unirradiated sensors and can be used to produce the first full scale\n\nsimulation of the ATLAS detector with the IBL that include sensor effects such as slim\n\nedge and thinning of the sensor.\n\nKeywords : Radiation Damage, Multi-Guard Ring Structure, Silicon detector, TCAD\n\nsimulation, Digitization, Planar Pixel Sensor, Slim edges, Test Beam, IBL, SLHC\n\n9\n\n\n\n\n\nTable des matières\n\nIntroduction 15\n\n1 The ATLAS experiment and upgrade project 17\n\n1.1 The Large Hadron Collider . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n\n1.2 The ATLAS experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n\n1.2.1 The Inner detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20\n\n1.2.2 The calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25\n\n1.2.3 The muon spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . 29\n\n1.3 THe ATLAS upgrade projects . . . . . . . . . . . . . . . . . . . . . . . . . . 30\n\n1.3.1 Phase 1 : The Insertable B-Layer (IBL) . . . . . . . . . . . . . . . . 32\n\n1.3.2 Phase 2 : Upgrade for high luminosity . . . . . . . . . . . . . . . . . 35\n\n2 Principles of Silicon pixel sensors 39\n\n2.1 The physics of Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40\n\n2.1.1 Semiconductors properties . . . . . . . . . . . . . . . . . . . . . . . . 40\n\n2.1.2 Charge transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42\n\n2.1.3 The pn junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43\n\n2.1.4 Physical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45\n\n2.2 Radiation detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52\n\n2.2.1 The energy deposition process . . . . . . . . . . . . . . . . . . . . . . 52\n\n11\n\n\n\nTABLE DES MATIÈRES\n\n2.2.2 Signal formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53\n\n2.3 The Hybrid Planar Pixel Sensor . . . . . . . . . . . . . . . . . . . . . . . . . 54\n\n2.4 Other Silicon sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56\n\n2.4.1 the 3D pixel sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58\n\n2.4.2 High Resistivity Monolithic Active Pixel Sensors (MAPS) . . . . . . 58\n\n2.5 Radiation damage in Silicon sensors . . . . . . . . . . . . . . . . . . . . . . 59\n\n2.5.1 Non-ionizing Energy Loss (NIEL) . . . . . . . . . . . . . . . . . . . . 59\n\n2.5.2 Ionizing energy loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66\n\n3 TCAD Simulation models 69\n\n3.1 Process simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71\n\n3.2 Device simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74\n\n3.2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74\n\n3.2.2 boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 75\n\n3.3 The Multi-Guard Ring structure . . . . . . . . . . . . . . . . . . . . . . . . 79\n\n3.3.1 Principles of guard ring structures . . . . . . . . . . . . . . . . . . . 81\n\n3.3.2 Optimization of guard ring structures for reduction of inactive area\n\nand radiation hardness . . . . . . . . . . . . . . . . . . . . . . . . . 84\n\n3.3.3 The Slim Edge Guard Ring structure . . . . . . . . . . . . . . . . . . 103\n\n3.4 The charge amplification mechanism in highly irradiated silicon sensors . . . 108\n\n4 From TCAD simulation to experimental data 117\n\n4.1 Experimental validation of TCAD simulation . . . . . . . . . . . . . . . . . 118\n\n4.1.1 Doping profile measurements . . . . . . . . . . . . . . . . . . . . . . 121\n\n4.1.2 Guard Ring measurements . . . . . . . . . . . . . . . . . . . . . . . . 136\n\n4.1.3 Current versus Bias characteristics . . . . . . . . . . . . . . . . . . . 141\n\n12\n\n\n\nTABLE DES MATIÈRES\n\n5 Planar Pixel Sensor digitization for ATLAS IBL simulation 149\n\n5.1 Test beam validation of TCAD simulation and digitization . . . . . . . . . . 152\n\n5.1.1 Validation of the digitization model . . . . . . . . . . . . . . . . . . . 153\n\n5.1.2 Edge effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157\n\nConclusion 163\n\nWord Cloud 170\n\nBibliography 170\n\n13\n\n\n\nTABLE DES MATIÈRES\n\n14\n\n\n\nIntroduction\n\nThe Large Hadron Collider, located in CERN, Geneva, has been delivering collisions\n\nof proton beam accelerated to an energy of 3.5 TeV since November 23 2009. since then,\n\nthe ATLAS experiment has been recording data to search for new physics to be discovered\n\nthrough analysis of the collision kinematic. The planned luminosity rise for the period\n\nbetween 2011 and 2020 will however bring new challenge to the detector that will need\n\nto be addressed to maintain the performance of the actual detector. The inner detector\n\nwill be the most challenged with the increased amount of tracks per collision to detect,\n\nresulting from the high luminosity upgrade. Radiation damage in its sensors and electronics\n\nwill eventually reduce the efficiency and lead to failure of the detector system. Aging of\n\nthe detector will also reduce the number of active channels and increase the detection\n\ninefficiency.\n\nAn intense R&D effort has been ongoing to design a new pixel detector capable of\n\nhandling increased occupancy linked to higher luminosity and able to resist to radiation\n\ndamage induced by the collisions. The Insertable B-Layer (IBL) project has been created to\n\nperform a first update of the detector by adding a new pixel layer between the beampipe\n\nand the actually inner layer of the pixel detector. The SLHC project plans to replace\n\nthe whole inner detector with an improved version able to withstand a ten fold increase\n\nin radiation damage and track multiplicity. In this work, I present a study, using TCAD\n\nsimulation, of the possible method of design of a planar pixel sensors reducing their inactive\n\narea and improving their radiation hardness. Different physical models available have been\n\nstudied to develop a coherent model of radiation damage in silicon that can be used to\n\npredict silicon pixel sensor behavior after exposure to radiation. The Multi-Guard Ring\n\nStructure used in pixel sensor design was studied to obtain guidelines for the reduction\n\n15\n\n\n\nINTRODUCTION\n\nof inactive edges detrimental to detector operation while keeping a good sensor behavior\n\nthrough its lifetime in the ATLAS detector. A campaign of measurement of the sensor’s\n\nprocess parameters and electrical behavior to validate and calibrate the TCAD simulation\n\nmodels and results will also be presented in this work. A model for charge collection in\n\nhighly irradiated diode was developed to explain the high charge collection observed in\n\nhighly irradiated devices.\n\nFinally , a simple planar pixel sensor digitization model to be used in test beam and\n\nfull detector system is presented. The simple model allow for easy comparison between\n\nexperimental data and prediction by the various radiation damage models available. The\n\ndigitizer has been validated using test beam data for unirradiated sensors and can be used\n\nto produce the first full scale simulation of the ATLAS detector with the IBL that include\n\nsensor effects such as slim edge and thinning of the sensor.\n\n16\n\n\n\nChapitre 1\n\nThe ATLAS experiment and upgrade\nproject\n\n1.1 The Large Hadron Collider\n\nThe Large Hadron Collider is a 27 km diameter proton accelerator and collider using\n\nsupra-conductive magnet technology, located in CERN, designed to operate at a nominal\n\nenergy of 7 TeV. The particle beam accelerated in the LHC originate from the accelerator\n\ncomplex present on site at CERN [1], as seen on figure 1.1. The proton synchrotron produce\n\na beam of an energy of 25 GeV that is the injected in the Super Proton synchrotron which\n\naccelerate the protons up to an energy of 450 GeV. The protons are then injected in the\n\nLHC and accelerated to an energy up to 7 TeV. The machine nominal operation luminosity\n\nis 1034 cm−2s−1. At the moment of writing these line, the LHC was operated at a beam\n\nenergy of 3.5 TeV for a peak luminosity of 8.3× 1032 cm−2s−1.\n\nFigure 1.2 show the peak luminosity recorded by the ATLAS detector and delivered by\n\nthe LHC for the period spanning from january 2011 to may 9th 2011. The luminosity is\n\nexponentially increasing toward the nominal value for which the machine was designed. A\n\nshutdown of the machine is planned for 2013 to allow maintenance and reparation. Work\n\non the magnet is scheduled to allow to reach nominal energy of 7 TeV per proton. Two\n\nbeam circulate in the clockwise and counterclockwise direction of the accelerator and can\n\ncollide at 4 main point around the ring, where are located the 4 main LHC experiments :\n\nATLAS, CMS, LHCb and ALICE.\n\n17\n\n\n\n1.1. THE LARGE HADRON COLLIDER\n\nFigure 1.1 – The LHC and CERN accelerator complex\n\nFigure 1.2 – ATLAS recorded Online peak luminosity per day for 2011\n\n18\n\n\n\n1.2. THE ATLAS EXPERIMENT\n\n1.2 The ATLAS experiment\n\nThe ATLAS (A Toroidal LHC ApparatuS) experiment is a particle collider installed on\n\nthe LHC experiment. This general purpose detector focus on the detection of the elusive\n\nHiggs boson through various decay channels and on the research of new physics beyond\n\nthe standard model . The ATLAS detector is composed of three main concentric sub-\n\ndetector represented on figure 1.3 : The inner detector, used for trajectory and impulsion\n\nmeasurement of charged particle, the calorimeters measuring the electromagnetic and ha-\n\ndronic energy deposition of the particles emanating from the interaction point and the\n\nmuon spectrometer used to measure the impulsion and trajectory of the weakly interacting\n\nmuons. The inner detector is enclose in a solenoid with a magnetic field of 2 T allowing\n\nthe measurement of the transverse momentum of the particle crossing its volume using the\n\ncurvature of the track in the magnetic field oriented along the beam direction. The muon\n\nspectrometer is enclosed in a toroidal magnet system with a magnetic field of 4 T.\n\nFigure 1.3 – The ATLAS detector\n\nThe ATLAS detector is designed to operate at a collision frequency of 40 MHz with\n\n19\n\n\n\n1.2. THE ATLAS EXPERIMENT\n\nbunch crossing every 25 ns. The amount of interaction at this rate and luminosity would\n\nbe to large to be recorded and a on detector trigger system. called level 1 trigger, filter\n\nthe interesting event using the calorimetric information. The trigger system identify the\n\nbunch crossing timestamp and retrieve from the detector the information related to that\n\ntimestamp from the detector systems holding the data. Higher level trigger system are then\n\napplied to further reduce the amount of data and select only interesting event emanating\n\nfrom the collision of the proton in the center of the machine.\n\n1.2.1 The Inner detector\n\nThe inner detector [2] is an essential part of ATLAS that has for purpose to measure\n\nthe trajectories of particles produced during the collisions of high energy protons beam\n\nproduced by the LHC. The detector is composed of three subsystems : The pixel detector ,\n\nthe Semi-Conductor Tracker (SCT) and the Transition radiation Tracker (TRT). The pixel\n\ndetector consists of three concentric barrels of pixel detector modules located at 5.05, 8.85\n\nand 12.25 cm from the interaction point and of 6 disk, 3 on each side of interaction point,\n\nlocated at a mean z of 49.5, 58.0 and 65.0 cm with regard to the interaction point. Fig. 1.4\n\nshows the cylindrical layers formed by the Inner Detector’s subsystems. Particle beams will\n\ntravel in the cylinder’s Z direction and collision will take place at the geometrical center of\n\nthe inner detector cylindrical structure.\n\nThe barrels are assembled with 1744 modules, shown in figure 1.5, each containing 16\n\nfront-end electronics 2880 channel chips, labeled FE-I3, bump-bonded to the pixel sensors\n\nmounted on a flex-hybrid circuit board, along with a module control chip (MCC). The\n\nmodule are assembled into ladders then paired to form a bistave structure, shown in figure\n\n1.6a. The bistaves are then assembled on the support tube to form half shell assemblies ,\n\nshown in figure 1.6b, that are then clamped together to form a pixel layer. The disk contain\n\neach 48 modules for a total of 288 modules for both end caps. The inner detector covers a\n\nregion of ±2.5η in the pseudo-rapidity coordinate system where η = −ln(tan(θ/2)). Each\n\nsensor exhibit pixels of 400x50 µm leading to a hit position reconstruction resolution of 12\n\nµm in φ at normal incidence.\n\nThe modules of the bistave assembly of the pixel barrels are overlapping in the Φ\n\n20\n\n\n\n1.2. THE ATLAS EXPERIMENT\n\nFigure 1.4 – 3D view of ATLAS Inner Detector, including Silicon tracker (SCT) and\nTransition Radiation Tracker (TRT) [3]\n\nFigure 1.5 – Schematic view of a barrel pixel module [2]\n\n21\n\n\n\n1.2. THE ATLAS EXPERIMENT\n\ndirection (angular coordinate in the cylindrical coordinate system), as shown on Fig. 1.7.\n\nThis overlap of detector modules is necessary to ensure an hermetic tracking coverage in Φ\n\nneeded to separate simultaneous tracks from high transverse energy particles produced by\n\nthe collisions at the interaction point. The edges of the pixel modules are not active area\n\nand need to be covered by the active area of the preceding and following modules along the\n\nhalf-shell assembly. This overlap of the sensors increase the amount of material between\n\nthe interaction point and the calorimeters and has been reduced as much as possible to\n\nlimit its effect on data quality.\n\n(a) The bistave assembly (b) Half shell assembly\n\nFigure 1.6 – Bistave and half-shell structure of the ATLAS pixel detector [4]\n\nThe position of the three detector layers close to the interaction point lead to a high\n\nnumber of charged and neutral particles crossing the detector. Radiation damage in these\n\nzone near the origin of the interactions will affect the performance of the innermost detector\n\nsystem. The inner detector innermost pixel layer, the b-layer, will be exposed to fluences of\n\nthe order of 5×1015 neq/cm\n2) at the end of its lifetime. Figure 1.8 show the simulated level\n\nof radiation damage in equivalent to 1 MeV neutrons for the inner detector for a 107 seconds\n\nof operation at nominal luminosity. The pixel sensors forming the tracker will suffer from\n\nhigh fluences phenomena like the formation of a double Junction/double electric field peak\n\nand Space-Charge Sign Inversion (SCSI) [5; 6]. Radiation effects also include an increase\n\nin the bias potential required to fully deplete the pixel sensors as the acceptor-like traps\n\n22\n\n\n\n1.2. THE ATLAS EXPERIMENT\n\nFigure 1.7 – Layout of the barrel pixel modules, R-Phi plane view [3]\n\nconcentration in silicon increases with exposure to radiation. The sensors will be operatedat\n\nfull depletion voltage in order to measure significant charge signals from particles crossing\n\nthe sensors, as collected charge is reduced by the trapping in defects created by radiation.\n\nThe Semiconductor tracker, or SCT, shown in figure 1.9, surround the pixel detector.\n\nThe detector is composed of two layers of micro-strip silicon p-in-n sensors rotated by an\n\nangle of 40 mrad with regard to each other. Strip are 80 µm x 128 mm long with the 80\n\nµm pitch oriented in the φ direction. Four cylindrical layers are positioned at respective\n\nradius of 29.9, 37.1, 44.3 and 51.4 cm with regard to the interaction point at φ = 0. Nine\n\nend-caps complete the system. The end-caps are located on each side of the interaction\n\npoint at a distances between 85 and 272 cm of the interaction point in the z direction. The\n\nz and r positioning resolution of the SCT modules are respectively of 580 µm and 17 µm\n\nand cover a region of ±2.5η in pseudo-rapidity.\n\nThe final part of the inner detector is the Transition Radiation Tracker sitting at a\n\nradius between 55.4 and 108.2 cm of the interaction point. The system is composed of\n\n351000 tubes of 4mm diameter filled with a gas mixture of Xe, CO2 and O2. A gold plated\n\nwire travel through the middle of the straw to form a detection electrode. A bias voltage\n\nof 1530V is applied between the outer surface and the wire to form an electric field inside\n\n23\n\n\n\n1.2. THE ATLAS EXPERIMENT\n\nFigure 1.8 – Annual 1 MeV neutron equivalent fluences assuming 107 s at nominal lumi-\nnosity [7]\n\n24\n\n\n\n1.2. THE ATLAS EXPERIMENT\n\nFigure 1.9 – The inner detector layer structure\n\nthe tube where ions and electron can drift . Particle crossing the tube deposit energy by\n\nionizing the gas and creating free charge that drift toward the electrode. Electrons crossing\n\nthe tubes will also produce transition radiation photon that will yield additional signal\n\nwith regard to other charge particles, allowing identification of the electrons using a two\n\nthreshold system. Positioning resolution in the r-φ plane is 130 µm.\n\n1.2.2 The calorimeter\n\nThe calorimeter system, shown in figure 1.10 surrounds the inner detector and is used\n\nto accurately measure the energy of the particle and jets coming from collision interactions.\n\nThe calorimeter is divided into two main parts : The electromagnetic calorimeter and the\n\nhadronic calorimeter. The design of the system has been done to minimize the dead area\n\nin the coverage in the r-φ plane and maximize the uniformity of the response to energy\n\ndeposition.\n\nThe electromagnetic calorimeter is a sampling calorimeter using liquid argon as the\n\n25\n\n\n\n1.2. THE ATLAS EXPERIMENT\n\nFigure 1.10 – The ATLAS calorimeter system\n\nFigure 1.11 – The ATLAS calorimeter system\n\n26\n\n\n\n1.2. THE ATLAS EXPERIMENT\n\ndetection medium. The accordeon electrode geometry, shown in figure 1.11 allow to obtain a\n\ncomplete r-φ coverage and uniform material budget. The barrel part of the electromagnetic\n\ncalorimeter extend between ±3.2η while the endcaps cover the range of 3.2 < |η| < 4.9.\n\nThe barrel of the calorimeter is divided in three longitudinal section for |η| < 2.5. and two\n\nsegment for 2.5 < |η| < 3.2. A presampler calorimeter with fine granularity strips is present\n\nfor |η| < 1.8 to allow correction for energy loss in the inner detector , cryostats and service\n\nlocated before the calorimeter. The measured energy resolution of the electromagnetic\n\ncalorimeter has been found to be described by equation 1.1 where a is a stochastic term\n\nequal to 10% and b a constant equal to 0.17%.\n\nσ(E)\n\nE\n=\n\na√\n(E(GeV ))\n\n⊕ b (1.1)\n\nThe forward region of the calorimeter, located in 3.1 < |η| < 4.9, is called the For-\n\nward Calorimeter (FCAL) and also use liquid argon as a detection medium. The readout\n\nelectrode are cylinder parallel to the beam direction inserted in a copper matrix with a\n\ndistance of 250 µm between the electrodes and the wall of the matrix, as shown in figure\n\n1.12. A quartz spacer is used to maintain the distance between the electrode tube and the\n\ncopper matrix and liquid argon fills the gap between the two surfaces.\n\nThe hadronic calorimeter is divided into two subsystems : The Tile Calorimeter and\n\nthe Hadronic End-Cap plus Forward Calorimeter. The Tile Calorimeters extend for the\n\n|η| < 1.6 pseudorapidity region of the detector. It is composed of modules scintillating tiles\n\nstacked between Iron absorbers slabs, as shown on figure 1.13, . Each tile is read through\n\nan wavelength shifting optical fibers coupled to photomultipliers. Timing resolution of the\n\nphotomultipliers is 23.5 ns , allowing to tag hadronic interaction to the correct Level 1\n\ntrigger clock.\n\nThe Forward Calorimeters and Hadronic End-Cap are liquid Argon sampling calorime-\n\nters located at each end of the detector, in the 3.1 < |η| < 4.9 pseudo-rapidity region. The\n\nHadronic End-Cap is formed of 128 modules of parallel plate liquid argon chambers , 32\n\nfor each wheel, an example of which can be seen in figure 1.14. The FCAL use the same\n\nelectrode structure as the electromagnetic calorimeter with a Tungsten matrix instead of\n\n27\n\n\n\n1.2. THE ATLAS EXPERIMENT\n\n(a) FCAL endcap copper matrix (b) FCAL endcap electrode geometry\n\nFigure 1.12 – The ATLAS forward calorimeter electrode structure and its copper matrix\n\nFigure 1.13 – The ATLAS Tile Calorimeter module\n\n28\n\n\n\n1.2. THE ATLAS EXPERIMENT\n\ncopper. Disposition of the different forward calorimeter system is described in figure 1.15.\n\nFigure 1.14 – An ATLAS End-Cap calorimeter wheel\n\nFigure 1.15 – ATLAS Forward region disposition of calorimeters systems\n\n1.2.3 The muon spectrometer\n\nThe muon spectrometer, shown in figure 1.16, is of a rapid detection system located\n\noutside the toroidal magnet system, used for triggering on muons coming from interactions,\n\nand a precision measurement system located inside the toroid magnetic field and used to\n\nmeasure accurately the impulsion of the muons by measuring the curvature of their tracks\n\nin the field. Three type of detection module are used in the detector system :\n\nThe Thin-Gap chamber (TGC)\n\n29\n\n\n\n1.3. THE ATLAS UPGRADE PROJECTS\n\nThe Cathode strip chambers (CSC)\n\nThe Monitored drift tubes (MDT)\n\nThe Resistive-plate chambers (RPC)\n\nThe precision measurement chambers are located in the central region of the detector\n\nat a radius of 5, 7.5 and 10 m with regard to the interaction point, and in the End-Cap\n\nregions at z = 7 and 22 m. They are composed of MDT and CSC modules. the fast trigger\n\nchambers are present in the |η| pseudo-rapidity region. They are composed of RPC for the\n\nbarrel section and TGC for the end caps.\n\nFigure 1.16 – The ATLAS Muon Spectrometer\n\nThe ATLAS detector has been recording data since november 23 2009. The readiness of\n\nall the subsystem has allowed a rapid progress in energy, luminosity and physics analysis.\n\nFigure 1.17 show an example of a Z muon candidate in the 2010 run data decaying into\n\n2 back to back electrons with transverse energy of 40 and 45 GeV. The electron track\n\nreconstruction in the inner detector is shown, along with their measured energy in the\n\nelectromagnetic calorimeter.\n\n1.3 THe ATLAS upgrade projects\n\nThe LHC was designed to produce a luminosity of 1034cm−2s−1. An upgrade of the\n\naccelerator is planned to increase even further the luminosity delivered by the machine.\n\nThree phase have been planned in the upgrade project [8]. Phase 1 will consist of an\n\n30\n\n\n\n1.3. THE ATLAS UPGRADE PROJECTS\n\nFigure 1.17 – Z boson candidate in the ATLAS detector\n\nincrease to a luminosity up to 2.3×1034cm−2s−1 without any modifications to the machine\n\nitself by pushing the machine to its maximum performance. Phase 2, called the SLHC\n\nphase, would see a factor 10 increase in luminosity with regard to the nominal luminosity.\n\nModification to the injection system will be performed to increase injection energy in the\n\nLHC at 1TeV, instead of the 450 GeV currently produced by the SPS. Modification to\n\nthe insertion quadrupole and to the machine parameters would then allow to increase\n\nluminosity.\n\nThe ATLAS pixel detector, located very close to interaction point, will see a increasing\n\namount of radiation as a consequence of the luminosity upgrade. The system was designed\n\nto operate at the actual nominal luminosity and modification to the system will be needed\n\nto cope with the increased pile-up of interaction at each bunch crossing to ensure a stable\n\ndetection efficiency, independent of luminosity. Two upgrade projects are planned to add\n\na new pixel layer and eventually replace the entire pixel detector to handle the problems\n\nbrought by an increase in luminosity : The Insertable B-Layer (IBL) and the SLHC upgrade.\n\n31\n\n\n\n1.3. THE ATLAS UPGRADE PROJECTS\n\n1.3.1 Phase 1 : The Insertable B-Layer (IBL)\n\nThe IBL project [9] consist in building a new layer for the pixel detector that will be\n\ninserted between the beampipe and the actual b-layer. The main motivation behind this\n\ndetector upgrade are :\n\n1. Tracking Robustness : Failure of actual module as they age and suffer from radiation\n\ndamage will affect tracking efficiency and precision. The addition of an additional\n\nspace point will provide a more robust tracking by compensating for the failed sensors\n\nand by giving information about tracks from a distance closer to interaction point.\n\n2. Luminosity effects : Increase in luminosity will produce more tracks per bunch cros-\n\nsing to be reconstructed by the pixel detector. Readout inefficiencies on the current\n\nsensor will reduce the overall detection inefficiency and the IBL will provide a system\n\nwith a increased occupancy capacity to compensate for this effect.\n\n3. Tracking precision : The location of the IBL, closer to interaction point, will allow\n\nfor more accurate reconstruction of the primary vertex and b-tagging. The insertion\n\nof the IBL will result in an increase in sensitivity to physics channel using these\n\nparameters.\n\n4. Radiation Damage : The actual b-layer is foreseen to resist to radiation doses of 1×\n\n1015neqcm\n−2. In the best scenario of luminosity upgrade, radiation damage inflicted\n\nduring pixel detector operation could eventually lead to a failure of the system due\n\nto radiation damage. The insertion of the IBL will provide a insurance policy against\n\nsuch problems if they were to occur.\n\nThe main parameters of the IBL are described in table 1.1. Two sensor candidate are\n\ncurrently considered to build the IBL : Planar Pixel Sensor and 3D Silicon Sensors. The\n\ntwo technology require different module design, with a single chip module being used for\n\n3D sensors while a double chip module is to be used with Planar sensors. A new readout\n\nchip has been developed to replace the FE-I3 used in the current ATLAS pixel modules.\n\nTable 1.2 show a comparison of the main characteristics of the two readout chip. The\n\nFEI4 can be distinguished by it larger size, smaller pixels, increased radiation hardness\n\nand occupancy performance. The timing resolution is kept at the same level as before but\n\n32\n\n\n\n1.3. THE ATLAS UPGRADE PROJECTS\n\nthe energy deposition measurement resolution is reduced from 8 to 4 bits. The reduction\n\nof the resolution was done to allow higher occupancy in the readout chip.\n\nTable 1.1 – Main IBL layout parameters [9]\n\nParameter Value\nNumber of staves 14\n\nNumber of modules per stave (single/double FEI4) 32/16\nPixel size (φ,z) 50,250 µm\n\nModule active size WxL (single/double FEI4 modules) 16.8 x 40.8/20.4 mm2\n\nCoverage in η |η| < 3.0\n\nOverlap in φ between staves 1.82 degree\nCenter of sensor radius 33.25 mm\n\nSensor thickness 150-250 µm\nRadiation length at z=0 1.54% of X0\n\nTable 1.2 – Comparison of FE-I4 and FE-I3 readout chip\n\nParameter FE-I4 FE-I3\nTechnology IBM 130nm IBM 250nm\n\nPixel size (µm) 250x50 400x50\nNr. of Channel 26880 2880\n\nMatrix dimension 80 column sx 336 rows 18 columns x 164 rows\nhline Time-Over-Threshold resolution 4 bits 8 bits\n\nTiming resolution 25 ns 25 ns\nHit Buffer 32 per 4 pixels 64 per column pair\n\nThe IBL will be positioned at 3.325 cm of the interaction point and it will be required\n\nto replace the actual beampipe , located at a radius of 2,9 cm, by a new beampipe with a\n\nradius of 2,5 cm to liberate space to insert the IBL in the ATLAS pixel detector. Figure\n\nThe planar pixel sensor candidate for IBL was developed within the Planar Pixel Up-\n\ngrade Group. Three model were developed, using recommendation from the simulation\n\nwork presented in this thesis :\n\n– Conservative n-in-n : ATLAS Standard sensor with a reduction of the guard ring by\n\nremoval of the 3 outer guard ring to obtain the edge width of 450 µm required for\n\nIBL\n\n33\n\n\n\n1.3. THE ATLAS UPGRADE PROJECTS\n\nFigure 1.18 – IBL r − φ layout view\n\n(a) IBL material budget (b) FInner detector material budget (IBL included)\n\nFigure 1.19 – Radiation length as a function of η in the IBL and inner detector\n\n34\n\n\n\n1.3. THE ATLAS UPGRADE PROJECTS\n\n– Slim Edge n-in-n ATLAS sensor : ATLAS Standard sensor with 3 guard ring removed\n\nand guard ring shifted under the pixels to obtain an inactive edge of 100 µm.\n\n– n-in-p sensor : Planar pixel sensor using a p substrate and a 450 µm reduced guard\n\nring structure\n\nFigure 1.20 shows the edge geometry for the three candidate. More details on the\n\nstructure, their behavior and design will be given in the next chapters.\n\n(a) Conservative n-in-n sensor (b) Slim edge n-in-n sensor (c) N-in-p sensor\n\nFigure 1.20 – Planar pixel sensor candidates for IBL\n\n1.3.2 Phase 2 : Upgrade for high luminosity\n\nThe phase two of LHC upgrade will require a completely new inner detector to survive\n\nluminosity and pile-up foreseen in LHC. The amount of radiation damage on the pixel\n\nsensors will increase by a factor 10 with regard to the current pixel detector which will not\n\nbe able to cope with such occupancy and radiation damage. Research and development is\n\nnow ongoing to design a new layout for the ATLAS super-LHC inner detector. Figure 1.21\n\nshows the current accepted layout planned for the upgrade. The TRT is removed along\n\nwith the whole actual inner detector and replaced by 4 layers of pixels, 3 layers of short\n\nstrips and 2 layers of long strips [10].\n\nPlanar pixel sensor could be used for the SLHC pixel detector if it can yield enough\n\nsignal at the 1× 1016neqcm\n−2 fluence expected for the inner layer of the pixel detector. In\n\nchapter 3, a study of a new physical phenomenon observed in highly irradiated conditions\n\nhas been performed to understand the physics mechanism behind the formation of collected\n\nsignal in highly irradiated silicon sensors. The charge amplification can be explained by\n\nimpact ionization current and trap-to-band tunneling in the bulk of the sensor and could\n\n35\n\n\n\n1.3. THE ATLAS UPGRADE PROJECTS\n\nFigure 1.21 – ATLAS super-LHC inner detector planned z − yi layout view\n\n36\n\n\n\n1.3. THE ATLAS UPGRADE PROJECTS\n\nbe used to design a new sensor able to survive the harsh environment of the SLHC ATLAS\n\ndetector. The R&D done in the framework of the IBL upgrade also covers the needs of the\n\nSLHC detector and results presented in this work can also be applied to detectors in the\n\nSLHC environment,\n\n37\n\n\n\n1.3. THE ATLAS UPGRADE PROJECTS\n\n38\n\n\n\nChapitre 2\n\nPrinciples of Silicon pixel sensors\n\nSilicon-based detector have been used for the last 60 years as an efficient mean to\n\ndetect the presence of charged particles. Gold contact barrier, then p-n junction diodes\n\nwere used between 1955 and 1965 as an efficient small size spectroscopic sensor to measure\n\nthe ionizing energy deposition of β particles in silicon. The first HEP experiment to make\n\na wide use of silicon as a tracking detector was CERN’s NA11 and NA32 experiments [11].\n\nThe strip sensors used in their tracking system shown the possibility of large scale usage\n\nof these sensors in tracking applications in HEP.\n\nNowadays, the wide use of silicon diode based sensors allowed for the development of\n\nvery sophisticated detectors used in large scale experiments where high resolution tracking\n\nof charged particles is required. Their increased exposure to radiation in high luminosity\n\nexperiments due to their proximity to the interaction points remains a challenge as aging\n\neffects have been observed with increasing exposure to high energy particles. In this chapter\n\nI will describe the basic semiconductor physics behind the p-n junction based detectors and\n\nthe mechanism that allow them to be used as efficient radiation detection devices. I will\n\ndescribe the ATLAS Hybrid planar pixel sensor which is the main study subject of this\n\nthesis. A quick survey of the concurrent technologies will be presented.\n\nFinally, the physics behind the aging effects of silicon detectors due to exposure to\n\nradiation will be described. The empirical models developed in the recent years will be\n\ncompared to the microscopic effects of radiation damage of silicon to provide an explanation\n\nof the behavior observed in irradiated silicon sensors.\n\n39\n\n\n\n2.1. THE PHYSICS OF SILICON\n\n2.1 The physics of Silicon\n\n2.1.1 Semiconductors properties\n\nSilicon used in particle detection is a semiconductor with a face-centered cubic crystal-\n\nline structure, as shown in figure 2.1.\n\nFigure 2.1 – Unit cell of silicon crystalline structure\n\nThe periodic nature of the silicon crystal lattice create the conditions for the electrons\n\nof the silicon atoms to arrange, in the energy-impulsion domain, in a band structure with\n\nforbidden zones where an electron cannot be found. This energy region where electrons\n\ncannot be found is called the bandgap. The height of this region is a unique property of\n\neach semiconductor.\n\nThe valence electrons of the silicons atoms are distributed between the conduction band\n\nand the valence band. The electrons in the conduction band are weakly bound to the lattice\n\nand can freely move through the material while the valence electrons are constrained to\n\nstay close the lattice structure. The Pauli exclusion principle forbids that two fermions,\n\nsuch as electrons, lie in the same energy-impulsion state. The energy distribution of this\n\ngas of fermions can be best described by the Fermi-Dirac distribution function (eq. 2.1) :\n\nni =\ngi\n\ne\nEi−µ\nkBT + 1\n\n(2.1)\n\nwhere gi is the degeneracy factor of the energy state and µ is the Fermi-Dirac quasi-energy\n\n40\n\n\n\n2.1. THE PHYSICS OF SILICON\n\nlevel. At low temperature (kBT << µ), µ is the energy level under which all state are\n\noccupied. At high temperature (kBT >> µ),µ is the energy level over which a state has\n\nonly a probability of 50% of being occupied. The Fermi-Dirac distribution at different\n\ntemperature is shown in figure 2.2.\n\nFigure 2.2 – Fermi-Dirac Distribution, µ = 0, gi = 1\n\nThis principle leads to the introduction of a virtual particle, the hole, as the carrier\n\nof the \"absence of an electron\" in the valence band of silicon. The addition of impurities\n\nto the crystalline structure modify the macroscopic properties of silicon. When impurities\n\ninserted in the crystal structure have a number of valence electrons (Nve) different than\n\n4, for silicon, extra electrons (Nve > 4) or holes (Nve < 4) are introduced in the band\n\nstructure. The hole introducing dopant is called an acceptor and is found in p type silicon.\n\nThe electron introducing impurities are called donors and are found in n type silicon.\n\nBoth type of dopant can be present in silicon and will compensate each other in such\n\nway that only the net dopant concentration will be electrically active. In n type and p\n\ntype semiconductors , the Fermi-Dirac quasi-energy level is found respectively close to the\n\nconduction or the valence band. The additional carriers will fill energy states not occupied\n\nin the intrinsic silicon modifying the position of the Fermi-Dirac quasi-energy level. At high\n\n41\n\n\n\n2.1. THE PHYSICS OF SILICON\n\ntemperature typical of the operation of silicon sensors, most of the impurities additional\n\ncarriers are added to the cloud of free carriers in the silicon crystalline structure, increasing\n\nthe amount of carriers available for conduction. In highly doped silicon (ND,A > 1018cm−3),\n\nthe amount of additional carriers is such that energy levels in the valence band, for holes,\n\nor in the conduction band, for electrons, have a probability close or equal to 1 of being\n\noccupied. This lead to high conductivity of the material even at low temperature.\n\n2.1.2 Charge transport\n\nThe dynamics of the carriers inside silicon can be described by the drift-diffusion equa-\n\ntions (eq. 2.2 and 2.3) coupled to the Poisson (eq. 2.4) [12] :\n\ndp\n\ndt\n= ∇ ·Dh∇p+∇ · (pµh ~E) +Gh −Rh (2.2)\n\ndn\n\ndt\n= ∇ ·De∇n−∇ · (nµe ~E) +Ge −Re (2.3)\n\n−∇2V = ∇ · ~E =\nρ\n\nε\n(2.4)\n\nwhere p and n are respectively the density of holes and electrons in [ 1\ncm3 ], D in [ cm\n\n2\n\ns ],\n\ntheir respective diffusion coefficient, µ the mobility in [cm2/V/s]. G is the generation rate\n\nand R, the recombination rate, both in [1/cm3/s]. The h and e subscript respectively design\n\nholes and electrons. ρ is the net charge density in [C/cm3], where C are Coulomb.\n\nGeneration/Recombination terms are important to describe the behavior of silicon de-\n\ntectors. Generation is responsible for leakage current present in detectors under bias. Re-\n\ncombination occurs between free carriers and its rate is proportional to the concentration\n\nof the most rare carrier. Silicon being an indirect gap semiconductor, generation and re-\n\ncombination occurs mostly through the defect states that are present in the bandgap of\n\nSilicon.\n\n42\n\n\n\n2.1. THE PHYSICS OF SILICON\n\n2.1.3 The pn junction\n\nWhen two regions of silicon containing different concentration of free carriers are put\n\ninto contact, we form a junction if the two region are respectively of p and n type. At the\n\ncontact region, the excess holes and electrons present create an electric field dragging free\n\ncarriers on the other side of the junction . A space-charge region is created at the junction\n\nwhere the electric field is present. The rate of carriers entering the space-charge region\n\nby diffusion from the doped region is equal to the rate of carriers leaving this region by\n\ndrifting in the electrical field built by the difference of carrier concentration on each side\n\nof the junction. A zone with very low concentration of free carriers that is created at the\n\njunction is called the depletion zone. The only free carriers present are coming from the\n\nthermally generated carriers created to replace the carriers drifting away in the electric\n\nfield. The application of an additional electric field through electrodes in contact with each\n\nside of the junction will modify this equilibrium by dragging carriers from the doped regions\n\naway or within the junction and modify the width of the depletion zone.\n\nThe pn junction can be biased in two different manners, as illustrated in figure 2.3.\n\nIf a negative voltage is applied on the n side and a positive voltage on the p side of the\n\njunction, the depletion zone tends to shrink and the electric field in the depletion zone is\n\nreduced, allowing more charge to cross the potential barrier and diffuse in the opposite side\n\nof the junction. This results in a reduction of the apparent resistance of the pn junction\n\nand a non-linear exponentially increasing current. In the opposite case , the depletion\n\nregion will expand away from the junction and the electric field increases as the amount\n\nof space charge becomes more important. The only current flowing through the junction is\n\nthe diffusion current from each depletion region edges, which saturates with distance, and\n\nthe generation/recombination current from the depletion region.This result in an apparent\n\nincrease of the resistance of the pn junction and results in a saturation of leakage current.\n\nFigure 4.21 show the typical Current versus bias of a pn junction.\n\nThe depleted zone in reverse bias mode represent the volume where particle energy\n\ndeposition can be detected as carriers generated through this process can drift into the\n\nelectric field and generate a signal. The width of the depletion zone can be calculated using\n\n43\n\n\n\n2.1. THE PHYSICS OF SILICON\n\nFigure 2.3 – The three possible state of a pn junction : at rest (top), forward bias (middle),\nreverse bias (bottom). The depletion region relative size is shown on the figure with the\nspace-charge sign of each zone of the diode.\n\nequation 2.5 [13].\n\nd =\n\n√\n2εV\n\neN\n(2.5)\n\nWhere ε is the electrical permittivity in silicon, V the applied bias voltage, e the electron\n\ncharge and N the dopant concentration in the region where dopants are less concentrated.\n\nDopant concentration can be linked to sensor resistivity using equation 2.6, details of the\n\ncalculation are given in the annex.\n\nρ =\n1\n\neND,Aµn, p\n(2.6)\n\n44\n\n\n\n2.1. THE PHYSICS OF SILICON\n\nFigure 2.4 – Typical pn junction current versus bias characteristics.\n\nWhere ND,A is the donor or acceptor effective concentration.\n\n2.1.4 Physical models\n\nIn order to simulate correctly the behavior of silicon detectors, the generation and\n\nrecombination terms of the drift-diffusion equations, mobilities of the carriers, oxide-silicon\n\ninterface and metal silicon interface physics must be correctly modeled. The dependence of\n\nmodels on electric field magnitude, temperature, dopant concentration must be taken into\n\naccount correctly to obtain an accurate and quantitative simulation of a device.\n\n2.1.4.1 Generation-Recombination\n\nThe rates of generation and recombination of thermally generated carriers are descri-\n\nbed by the modified Shockley-Read-Hall equation ([12]), which describes the generation-\n\nrecombination in indirect-band gap semiconductors such as silicon. This model assumes\n\nthat the transition of carriers between bands occurs through a single trap energy level\n\n45\n\n\n\n2.1. THE PHYSICS OF SILICON\n\nlocated deeply in the gap, Etrap.\n\nRSRH =\npn− n2\n\ni\n\nτp[n+ nie\nEtrap\nkbT ]τn[p+ nie\n\n−Etrap\nkbT ]\n\n(2.7)\n\nτn =\nτn0\n\n1 +\nNdopant\nNSRHn\n\n(2.8)\n\nτp =\nτp0\n\n1 +\nNdopant\nNSRHp\n\n(2.9)\n\nEquation (2.7) gives the Concentration-Dependent Shockley-Read-Hall Generation-\n\nRecombination model used in our simulation, where (2.8) and (2.9) explicit the concen-\n\ntration dependence. τp,n are the recombination lifetime for holes and electrons, τn0, τp0 =\n\n10−5 s a material dependent parameter representing the recombination lifetime for low\n\ndopant concentration bulk, NSRHp,n = 5× 1016 cm−3 a material dependent empirical pa-\n\nrameters and Ndopant the dopant concentration. The user dependent parameters must be\n\nchosen to represent the bulk material simulated. The values presented here were selected\n\nto represent a typical high resistivity bulk used for particle detectors that are not pure\n\ncrystals. The presence of oxygen and other impurities affects its electrical properties. Do-\n\npant are also introduced during fabrication of the sensors whereas defects are introduced\n\nby high energy particles crossing the sensor. In the super-LHC environment , ATLAS inner\n\ntracker will be exposed to high level of radiation and the large introduction of structu-\n\nral defects must be taken into account in the design of the sensors. More sophisticated\n\nsimulations of bulk properties like leakage current requires a more complex description of\n\ngeneration-recombination mechanisms.\n\nOur simulation of irradiated sensors use the modified Shockley-Read-Hall Generation-\n\nRecombination model, which can take into account the presence of multiple trap levels in\n\nthe band gap, introduced by radiation or native defects. Generation-Recombination terms\n\nfor each trap are calculated using (2.7) and a global term Rtotal is calculated following\n\n46\n\n\n\n2.1. THE PHYSICS OF SILICON\n\n(2.10).\n\nRtotal =\nl∑\n\nα=1\n\nRDα +\nm∑\nβ=1\n\nRAβ (2.10)\n\nτn,p =\n1\n\nNtvth,n,pσn,p\n(2.11)\n\nl and m are the numbers of donors and acceptors traps,RA,β RD,α the Generation-\n\nRecombination terms for respectively acceptors and donors traps. The density of traps Nt\n\nis taken into account through the parameters τn and τp used for each trap level, as shown\n\nin (2.11) .\n\nFinally, charge states of traps are taken into account in Poisson equation right term.\n\nThe amount of ionized trap is determined using Boltzmann statistics. This complex model\n\ntake into account the variation of the effective doping density and temperature depen-\n\ndence and model correctly the contribution of thermally generated carriers of generation-\n\nrecombination term of the transport equations. However, the presence of an intense electric\n\nfield (O(100kV cm−1)) alter the bandgap structure of silicon and enhance the generation\n\nand recombination of carriers.\n\nThe electric field presence causes a bending of the bandgap structure in space, lowering\n\nthe potential barrier faced by carriers to cross to or from the traps present in the bulk\n\nmaterial, as shown in figure 2.5.\n\nThis affects the lifetime of electrons and holes trapped in the defects in the band gap\n\nof silicon. An increased electric field in the bulk of the sensor will bend the band gap of\n\nsilicon modifying the energy level of the conduction and valence band between different\n\nspace points. If this bending is sufficient, tunneling of the carriers trapped in the defects\n\nto the valence or conduction band can occur, reducing the effective lifetime of the trapped\n\ncarrier and contributing to the leakage current and thus to the generation rate term of\n\nthe drift-diffusion equations. This physical phenomenon is called trap-to-band phonon-\n\nassisted tunneling. A large amount of defects are present in irradiated silicon and high\n\nvoltage operation is needed to obtain charge collection recovery. A model elaborated and\n\nintroduced in our simulation is described in details in [14]. In this model, trap lifetime of\n\n47\n\n\n\n2.1. THE PHYSICS OF SILICON\n\nFigure 2.5 – Schematics of bandgap bending due to electric field and how it enhance the\ntunneling between traps and the bands.\n\nthe generation recombination terms of equation 2.8 and 2.9 is modified following equation\n\n2.12\n\nτp,n =\nτ0\np,n\n\n1 + ΓDiracp,n\n\n(2.12)\n\nΓDiracp,n =\n∆Ep,n\nkbT\n\n∫ 1\n\n0\ne\n\n(\n∆Ep,n\nkbT\n\n−Kp,nu3/2)\ndu (2.13)\n\nKp,n =\n4\n\n3\n\n√\n2m0m∗∆E3\n\np,n\n\n2q~E\n(2.14)\n\nWhere τ0\np,n is the trap lifetime without electric field, ∆Ep,n, the trap to band energy dif-\n\nference, m0 the effective carrier mass, m∗ = 0.15 the effective tunneling mass of the carrier,\n\nq the elementary charge, ~ the planck constant and E the local electric field magnitude.\n\nFigure 2.6 shows the typical trapped charge lifetime dependence on electric field fol-\n\nlowing the model introduced to the simulation. As the time scale of the charge drift in\n\nsilicon sensors is O(10 ns) this effect can be important for the charge collection in irra-\n\ndiated sensors operated at high bias voltage. The effect of trap-to-band tunneling is taken\n\ninto account in transient simulation by affecting the terms of equation 2.29\n\n48\n\n\n\n2.1. THE PHYSICS OF SILICON\n\nFigure 2.6 – Trapped charge lifetime dependence on electric field, Etrap = Ec − 0.53 eV\n\nThe presence of an intense electric field in the bulk of silicon will lead to acceleration\n\nof carriers and generation of additional electron hole pairs by energy transfer from the\n\naccelerated carrier to the lattice electrons by coulombian scattering. This effects lead to\n\nbreakdown of silicon sensors and must be modeled to reproduce this behavior in simulation.\n\nA model of impact ionization has also been implemented in the generation rate of the drift-\n\ndiffusion equation. This model is the Selberherr Impact ionization model [15]. Equation\n\n(2.15) shows the relation used to obtain the generation rate contribution from impact\n\nionization, with the electric field dependence detailed in equation (2.16).\n\nGimpact = αn( ~E)\n∣∣∣ ~Jn∣∣∣+ αp( ~E)\n\n∣∣∣ ~Jp∣∣∣ (2.15)\n\nαn,p( ~E) = An,pe\n−Bn,p\n\n~E (2.16)\n\nCoefficients An,p and Bn,p are determined experimentally and are chosen as a function of\n\nthe material. Fig. 2.7 shows the electric field dependence of the impact ionization coefficient\n\nαn. If the electric field inside the irradiated sensor bulk reaches a magnitude of the order\n\nof 100 kV cm−1, some multiplication effects leading to increased leakage current are to be\n\nexpected.\n\n49\n\n\n\n2.1. THE PHYSICS OF SILICON\n\nFigure 2.7 – Impact generation coefficient αn dependence on electric field magnitude\n\n2.1.4.2 Mobility\n\nMobility of charge carriers in silicon is influenced by the magnitude of the parallel elec-\n\ntric field in which it is drifting. In a high electric field, free carrier’s energy loss by inelastic\n\nscattering in the crystal lattice will balance with the energy gained from acceleration in\n\nthe electric field. This leads to the saturation of the carrier’s speed [12]. This effect can be\n\nexpressed in terms of a variation of the mobility (µ) as a function of the parallel electric\n\nfield magnitude [16]. Equation 2.17 and 2.18 show the typical expression used to model the\n\nparallel field dependence of mobility in Silicon.\n\nµ(E) = µ0(\n1\n\n1 + (µ0E\nvsat\n\n)β\n)−β (2.17)\n\nvsat =\nα\n\n1 + θe\nTL\n\nTnominal\n\n(2.18)\n\nWhere TL is the lattice temperature. Typical values used for this model are shown in\n\n50\n\n\n\n2.1. THE PHYSICS OF SILICON\n\ntable 2.1.\n\nTable 2.1 – Parallel field dependence mobility model parameters\n\nParameter Electrons holes\nµ0 (cm2/V s at 300K) 1400 450\n\nβ 2.0 1.0\nα (cm/s) 2.4× 10−7 2.4× 10−7\n\nθ 0.8 0.8\nTnominal (K) 600 600\n\n2.1.4.3 Magnetic field effects\n\nSemiconductors sensors for particle tracking application are often used in a magnetic\n\nfield to allow the measurement of the transverse momentum of incoming particles. While\n\nthe presence of a field modify the behavior of the detectors, they can be operated in intense\n\nmagnetic field without hurting the detector performances. The carriers inside the magnetic\n\nfield are subject to the Lorentz force (equation 2.19), where is the velocity of the carrier\n\nand B the magnetic field intensity.\n\n~F = q( ~E + v × ~B) (2.19)\n\nThe additional component of the force due to the magnetic field presence leads to\n\ncarriers drifting away from the electric field lines with an angle determine by equation\n\n2.20 [17], where theta is the angle between the electric field and the actual carrier drift\n\ntrajectory and µHp,n is the Hall mobility, which differs from normal mobility because of the\n\neffects of the presence of the magnetic field.\n\ntanθp,n = µHp,nB (2.20)\n\n51\n\n\n\n2.2. RADIATION DETECTION\n\n2.2 Radiation detection\n\n2.2.1 The energy deposition process\n\nThe reverse bias operation mode of the silicon pn diode presents interesting characteris-\n\ntics for charged particle and x-ray detection. The radiation interacting with silicon diodes\n\nthrough ionizing process such as photoelectric effect, Compton scattering and through\n\ntransfer of energy to the bound carriers by coulombial scattering generate an amount of\n\nfree carriers in electron-hole pairs, proportional to the amount of energy deposited in the\n\ndiode through these ionizing processes. The mean amount of carriers generated in such\n\nprocess (Ne,h) is given by eq. 2.21 where Ed is the deposited energy and Eg is the pair\n\nproduction energy, 3.64eV for silicon.\n\nNe,h =\nEd\nEg\n\n(2.21)\n\nThe pair production process is an almost stochastic one, but the pair production process\n\nis not independent from pair to pair as the energy deposition occurs in a cascade of energy\n\ntransfers from the incoming particle to the carriers and from excited carriers to other\n\ncarriers. This correlation between the different pair production events gives rise to the\n\nFano factor (F=0.118 for silicon [18])in the intrinsic energy resolution equation (eq. 2.22)\n\nof semiconductor sensors to account for the lower standard deviation observed with regard\n\nto the Poisson predicted standard deviation.\n\nσSi =\n√\nFNe,h (2.22)\n\nIn a silicon diode used as a particle sensor, a diode is biased and the depleted re-\n\ngion represents the active detection region. A particle deposit its energy and a cloud of\n\nelectron-hole pairs are created. Following the electric field, they drift toward the electrodes\n\ngenerating an additional current in the diode that can be measured using detection elec-\n\ntronics, as shown in figure 2.8.\n\nThe low reverse biased diode leakage current represent an advantage of silicon sensors.\n\nThis leakage current is caused by electron-hole pairs being thermally generated in the the\n\n52\n\n\n\n2.2. RADIATION DETECTION\n\nFigure 2.8 – Schematics of ionizing particle detection in a reverse biased diode. The\nfree electron hole pairs produced by the particle energy loss drift in the electric field and\nproduce a current in the diode. Thermally generated (green boxes) are also generated in\nthe depleted zone of the diode.\n\ndepleted zone of the diode. These pairs also induce signal on the read-out electrode and\n\nare emitted randomly thus subject to statistical variations. This increases the noise in the\n\nsensor and low leakage current present in reverse-biased diodes makes it a perfect detection\n\nmedium for ionizing radiation.\n\n2.2.2 Signal formation\n\nThe signal induced on a readout electrode is not due to the collection of the free carriers\n\nthemselves. The charge’s electric field flux inside the readout electrode varies as the charge\n\ndrift into the sensor and displacement current is created and generate the detectable signal.\n\nThe real signal can be calculated using the Ramo theorem [19], as shown in 2.23.\n\nQk =\n∑\ni\n\nqiφk(~rifinal)−\n∑\ni\n\nqiφk(~ri0) (2.23)\n\nWhere Qk is the charge induced on electrode k, qi is the charge of the carrier i, ~ri its\n\n53\n\n\n\n2.3. THE HYBRID PLANAR PIXEL SENSOR\n\nposition and φk the Ramo potential of electrode k. The Ramo potential is calculated by\n\nsolving the Laplace’s equation in the geometry of the detector while imposing a Dirichlet\n\nboundary condition at the electrodes, with φ = 1 at the k’th electrode and 0 at the other\n\npresent electrode and a Neumann boundary condition (~∇φ · ~n = 0, ~n is the vector normal\n\nto the boundary) at the rest of the boundaries. This theorem can be applied to a variety\n\nof detection medium ranging from plasma to solid-state detectors. The presence of space\n\ncharge do not influence the calculation of the Ramo potential [20; 21]\n\n2.3 The Hybrid Planar Pixel Sensor\n\nThe planar pixel sensor (figure 2.9) consists of an array of small rectangular diodes\n\nbuilt by implanting dopant in a Silicon wafer to form junctions. A structure of guard rings\n\nis disposed around the array of pixels to insure a smooth transition of the surface bias\n\npotential from the active area to the edge of the sensor. Each of the concentric rings is\n\nself-biased with the inner ring taking the pixel or backside electrode potential and the outer\n\nring taking the edge potential, usually the same as the verso of the edge. Details on the\n\nguard ring structure will be given in chapter 3. Lithography methods are used to create\n\nthe individual pixels diodes and contact electrode. An implant and a metallization on the\n\nbackside of the wafer is created with a dopant type opposite of the pixel’s implant type.\n\nThis create an ohmic contact. Bias is applied between each side of the wafer to deplete the\n\nsensor and create an electric field in the bulk of the sensor allowing the drift of free carriers\n\ngenerated by ionizing particles. The carrier cloud generated is localized in a small region\n\naround the particle track allowing to measure the particle position in the sensor using\n\nsignal induced on the individual pixels. An integrated circuit, called a front-end, is also\n\nbuilt with individual cells of signal lecture and digitization electronics matching the pixels\n\non the planar sensor. The integrated circuits individual channels are coupled to the diodes\n\nusing the bump-bonding technique. Each channel can then be read individually in a digital\n\nformat to obtain the information on the position and energy of the detected particles. The\n\nassembly is then mounted on a PCB or a Flex-Hybrid containing the circuits to bias the\n\nsensors and the integrated circuits and readout and transmit the acquired data. The final\n\nassembly is call hybrid Planar pixel sensor module and can be used to form large system\n\n54\n\n\n\n2.3. THE HYBRID PLANAR PIXEL SENSOR\n\nof particle tracking used in large HEP experiments such as ATLAS.\n\nFigure 2.9 – Schematics of the hybrid planar pixel sensor.\n\nPlanar pixel sensors can be used in 4 configuration of implants and bulk with each\n\ntheir advantage and inconvenient. Figure 2.10 describe the possible geometries of pixel and\n\nguard ring that are suitable for pixel sensors.\n\nThe electrode readout implant type is chosen to select the main signal carrier of the\n\nsensor. N implant are used for readout of electrons which travel faster is silicon due to\n\ntheir higher mobility. They are favored for radiation hard sensor as they are less prone\n\nto trapping. Bulk type is chosen as a function of desired guard ring side and radiation\n\nhardness issues. As N type bulk invert to P type-like bulk and depletion voltage rise after\n\nirradiation due to space charge sign inversion, the depleted zone, which expands from the\n\npn junction side towards the other junction, could not be maintained for p-in-n and p-in-p\n\nsensors when depletion voltage gets to high for the power supply. For n-in-n sensors, space\n\ncharge sign inversion is actually beneficial as the detector is always depleted under the\n\nreadout electrodes. Before inversion, depletion voltage is reduced with regard to the initial\n\nvalue hence the sensor can always be fully depleted. N-in-p sensors are unaffected by this\n\n55\n\n\n\n2.4. OTHER SILICON SENSORS\n\n(a) n-in-n sensor (b) p-in-n sensor\n\n(c) n-in-p sensor (d) p-in-p sensor\n\nFigure 2.10 – Schematics of the possible pixel sensor implant and bulk configurations.\n\nas the depletion always occurs from the pixel side of the sensor. Finally, the guard ring,\n\nto be functional, must be located on the pn junction side of the sensor. In the n-in-p and\n\np-in-n sensor case, this mean some high voltage, coming from the backside, will be present\n\nin the guard ring region and might be detrimental to the chip also located on the readout\n\nelectrode side. However, in the case of these detector, all lithography to build the electrode\n\nstructure is located on the same side of the wafer, reducing the fabrication cost of the\n\ndetectors. Table 2.2 summarized the characteristics of each configuration.\n\nThe ATLAS pixel detector is built with n-in-n planar pixel sensors. This technology\n\nwas chosen for it many advantages and is the planar pixel sensor candidate technology for\n\nthe Insertable b-layer.\n\n2.4 Other Silicon sensors\n\nIn the recent years, new technologies have been developed to build pixel sensors for par-\n\nticle detection. The goal of these new technologies is to increase the signal formation speed,\n\nreduce inactive zones of the sensor, build thinner sensor and increase radiation hardness\n\n56\n\n\n\n2.4. OTHER SILICON SENSORS\n\nTable 2.2 – Summary of planar pixel sensor configurations\n\nSensor Type Advantages inconvenients\nn-in-n\n\n– electron signal\n– inversion of depletion direction\n\nafter SCSI\n– guard rings on backside\n\n– double-sided process\n\nn-in-p\n– electron signal\n– no SCSI\n– single-sided process\n\n– guard ring on pixel side\n\np-in-n\n– single-sided process – guard ring on pixel side\n\n– hole signal\n– hard to fully deplete after SCSI\n\np-in-p\n– no SCSI\n– guard rings on backside\n\n– hole signal\n– hard to deplete with increasing\n\nradiation\n– double-sided process\n\n57\n\n\n\n2.4. OTHER SILICON SENSORS\n\nfor applications where such characteristics are required. The two main new technologies\n\nare the 3D pixel sensor and the High Resistivity Monolithic Active Pixel Sensor (MAPS).\n\n2.4.1 the 3D pixel sensor\n\nThe main difference between the planar pixel sensors and the 3D variety is the orien-\n\ntation of the electrode implant in the wafer. Through chemical etching techniques, deep\n\nholes are created in the wafer and implants are created on the surface of the holes, forming\n\np-type and n-type columns in the wafer. They are then filled with a conductive material\n\nto form the anodes and the cathodes. The bias is applied between the two type of columns\n\nand drift of the carriers occurs laterally, as shown in figure 2.11. For sensors with a pixel\n\npitch smaller than the wafer thickness, this leads to a faster signal and smaller depletion\n\nvoltage. The possibility to bias literally avoid the problem of high voltage distribution at\n\nedges present in planar pixel sensors, discussed in chapter 3 of this thesis, allowing for small\n\ninactive edges. Finally , the short drift distances of the carriers reduce the signal loss due\n\nto trapping during long charge drift makes the 3D sensors more radiation hard by design,\n\nsince, as it will be discussed in the next section of this chapter, radiation damage induced\n\ncharge loss by trapping of carriers in discrete energy levels present in the band-gap of the\n\nsensor’s material.\n\n2.4.2 High Resistivity Monolithic Active Pixel Sensors (MAPS)\n\nMAPS sensors are built on a single high resistivity silicon wafer where a low resistivity\n\nsilicon layer have been grown by epitaxy. A small subsection of the pixel surface is occupied\n\nby a detection diode depleting in the high resistivity buried layer, while the rest of the\n\nsurface is occupied by CMOS electronics in the low resistivity bulk, forming the readout\n\nelectronics of the diode, as shown in figure 2.12. A small electric field is present in the\n\ndepleted region and a small signal is generated in the detection diode. The signal is then\n\namplified and digitize on the same wafer by the readout electronics. The presence of the\n\namplifier so close to the detection diode allows low noise operation even with the small\n\nsignal produced by the passage of a particle. The presence of the electronics on the same\n\nwafer used for detection eliminated the need for a front-end integrated circuits and the\n\n58\n\n\n\n2.5. RADIATION DAMAGE IN SILICON SENSORS\n\nFigure 2.11 – Schematics of the hybrid 3D pixel sensor.\n\nsmall thickness of the epitaxial layer and detection layer (down to 50 µm) allows to build\n\nvery thin sensors for low material budget applications.\n\n2.5 Radiation damage in Silicon sensors\n\nSilicon sensors can be damaged by the exposure to radiation. Several effects need to\n\nbe taken into account to design radiation hard sensors for use in harsh environment such\n\nas in ATLAS inner detector. Two kind of radiation damage are important in the case of\n\nsilicon sensors : Non-ionizing and ionizing energy loss by particles interacting with the\n\nsensor’s material. Each effect lead to specific changes in the sensors operation conditions\n\nand electrical characteristics.\n\n2.5.1 Non-ionizing Energy Loss (NIEL)\n\nExposure of planar pixel sensor to non-ionizing energy loss from protons, pions and\n\nneutrons modify its electrical properties in the following ways :\n\n– Space-Charge Sign Inversion (SCSI)\n\n59\n\n\n\n2.5. RADIATION DAMAGE IN SILICON SENSORS\n\nFigure 2.12 – Schematics of the High Resistivity Monolithic active pixel sensor (MAPS).\n\n– Modification of full depletion potential (Vfd) (high voltage operation after high-dose)\n\n– Increased trapping\n\n– Leakage current increase\n\nNIEL is usually expressed, for silicon sensors, in 1 MeV neutron-equivalent by square\n\ncentimeter, (neq/cm\n2). The energy loss by exposure to different particle types and energies\n\ncan be calculated by scaling the flux of the incoming particle on the sensor surface by the\n\nratio between the NIEL in the bulk for that particle and the NIEL for a 1 MeV neutron. A\n\nlarge number of publication measuring the NIEL scaling factor for various type of particle\n\nis available in the litterature.\n\nSCSI effect was first predicted then measured experimentally [5; 6]. It is expressed\n\nthrough an inversion of the space charge sign in the depleted region of silicon sensors. This\n\ninversion has been shown to be due to the introduction of electrically active defects in the\n\nbulk that compensate for the natively present defects. This leads to an apparent change of\n\nthe bulk type, changing the wafer side where the high electric field is present and eventually\n\nleading to a complex distribution of space charge in the bulk leading to the formation of\n\n60\n\n\n\n2.5. RADIATION DAMAGE IN SILICON SENSORS\n\na high field region on both side of the silicon wafer (N+ − p − n − p+ structures). From\n\nequation 2.5, we can observe that an apparent charge in the bulk’s acceptor or donor net\n\nquantity will also lead to a variation of the depletion voltage needed to fully deplete the\n\nsilicon sensor. The steady introduction of these electrically active defects will eventually\n\nlead to an increase of the need depletion voltage and may limit the capacity to deplete the\n\nsensor completely.\n\nSignal carriers are subject to recombination with the same probability as thermally\n\ngenerated carriers. The introduction of defects in the bulk will lead to a lower characteristic\n\nrecombination time of the free carriers through these defects. If this time is of the order, or\n\nlower than expected collection time, it will lead to a reduction of the collected charge with\n\nregard to the expected value before the introduction of the defects. This radiation damage\n\neffect will lead to lower detection efficiency, reduced resolution, lower signal to noise ratio\n\nand will force frequent recalibration of the sensor’s readout electronics.\n\nFinally, the traps in the band-gap are possible mediator for generation and recombina-\n\ntion of carriers in indirect band-gap semiconductors. The addition of new trap in the sensor\n\nmaterial will then lead to an increased probability of thermal emission of carriers leading\n\nto an increased leakage current. This represent an issue reducing the detector performance\n\nby increasing the noise and can lead to cooling problems and thermal runaway as the dissi-\n\npated energy becomes more important. Finally, if the current leaking through the sensor is\n\ntoo important for the front-end readout out current compensation circuit capacity , it can\n\nlead to non-linear behavior of the electronics reducing the detector overall performances.\n\nFor a trap t, introduced by radiation or present in the original detector material, four\n\nprocesses can be enhanced : hole capture (Rth), hole emission (Gth), electron capture (Rte)\n\nand electron emission (Gth). Equations 2.24,2.25,2.26 and 2.27 ([17]) show the rate of each\n\nthese processes as a function of the trap’s capture cross-section (σtp,n), thermal velocity\n\n(vth), trap density (Nt), free carrier concentration (p, n),hole or electron emission probabi-\n\nlity (εtp,n) and occupancy of the trap (Pt).\n\nRth = vth,pσ\nt\nppNtPt (2.24)\n\n61\n\n\n\n2.5. RADIATION DAMAGE IN SILICON SENSORS\n\nGth = vth,pε\nt\npNt(1− Pt) (2.25)\n\nRte = vth,nσ\nt\nnnNt(1− Pt) (2.26)\n\nGte = vth,nε\nt\nnNtPt (2.27)\n\nAt equilibrium, the sum all recombination and generation rates must equal 0 (equation\n\n2.28). In this state, some concentration of defects can remain charged and modify the space\n\ncharge distribution in the sensor, as seen in the SCSI effect. An increased trap density\n\nlead to bigger generation term and higher leakage current in reverse biased diode where\n\nrecombination terms are kept low due to low free carrier density.\n\n∑\nt\n\nGte,h =\n∑\nt\n\nRte,h (2.28)\n\nDuring collection time in the depleted zone of a reverse biased diode where carrier\n\ndensity is low, a quasi-static approximation, as shown in equation 2.29 can be used to\n\ndetermine the behavior of the signal’s carriers during charge collection. This model supposes\n\nno carriers exchange between the different defects present, which is a valid approximation\n\nif defect density is low. The localized higher carrier density (n, p) in the charge cloud\n\ngenerated by the energy deposition of a particle, recombination terms are enhanced and\n\nlead to trapping of the signal if a large density of trap is present. A radiation damage\n\nincrease, more trapping will occur and lead to lower charge collection efficiency.\n\ndn, p\n\ndt\n=\n∑\nt\n\nGte,h −\n∑\nt\n\nRte,h (2.29)\n\nFor a distribution of carriers drifting in a reverse biased diode, one can compute the\n\ncurrent induced on an electrode w using Ramo’s field (~Φw), as a function of genera-\n\n62\n\n\n\n2.5. RADIATION DAMAGE IN SILICON SENSORS\n\ntion/recombination rate, as shown in equation 2.30.\n\nIw(t) =\n\n∫\nvolume\n\nρ(~x,\n∑\nt\n\nRte,h(~x),\n∑\nt\n\nGte,h(~x))~∇Φwd~x (2.30)\n\nThis equation is however complex to solve and one can approximate the collected charge\n\nby supposing a uniform distribution of traps, a constant electrical field and a punctual\n\ncharge, neglecting generation of carriers. The recombination rate of the traps can be then\n\nexpressed in term of the trap characteristic lifetime (Re, h = 1/τ e,ht ). An average cha-\n\nracteristic lifetime can be calculated using equation 2.31 and the collected charge can be\n\nexpressed using equation 2.32.\n\n1\n\nτ\n=\n∑\nt\n\n1\n\nτt\n(2.31)\n\nQf =\n\n∫ tf\n\nt0\n\nQ0µe,h ~Ee\n−t/τ ~∇Φw(Q0µe,h ~Et)dt (2.32)\n\nThe hadronic interactions of particles with the atoms of the crystal lattice transfers to\n\nthem part of their kinetic energy and displaces them from their original position, creating\n\ndisorder in the crystalline structure. The displaced atoms can be moved to interstitial\n\nspace in the lattice, forming a defect, called interstitial defect, than can become electrically\n\nactive and modify the band structure of the silicon. The vacancies in the lattice left by the\n\nknocked-off atom can also create an electrically active defect called the vacancy defect.\n\nMoreover, the pre-exisisting defects and dopants in the silicon bulk can interact with\n\nthe radiation induced defects to form more complex hybrid defects with different electrical\n\nbehavior. The ROSE [22; 23; 24] and RD50 CERN collaboration [25; 26; 27; 28; 29] have\n\nworked to identify the defects that are important to understand the effects of non-ionizing\n\nradiation damage in silicon . The main important defects introduced in silicon by irradiation\n\ncan be found in table 2.3. The introduction rate of these different defects vary with the type\n\nand energy of the particle causing the damage, the present concentration of Oxygen in the\n\nbulk and of the thermal history of the silicon sample. Defect engineering can be performed\n\nby favoring the formation of non electrically active defects such as the (V −O2i) defect which\n\n63\n\n\n\n2.5. RADIATION DAMAGE IN SILICON SENSORS\n\nis enhanced by the presence of Oxygen dimers (O2i). Oxygenated high resistivity silicon\n\nexhibit a high concentration of such dimer and therefore can substain higher radiation doses\n\nbefore effects detrimental to the operation of the planar pixel sensor becomes significant.\n\nTable 2.3 – Important defects introduced by NIEL in silicon ([17; 27])\n\nDefect type Charge state Energy level (eV)\nInterstitial (I) I− EC − 0.39\n\nI0\n\nI− EV + 0.4\n\nVacancy (V) V −− EC − 0.09\nV − EC − 0.4\nV 0\n\nV + EV + 0.05\nV ++ EV + 0.13\n\nDivacancy (V2) V −−\n2 EC − 0.23\nV −\n\n2 EC − 0.39\nV 0\n\n2\n\nV +\n2 EV + 0.21\n\nA-Center (V-O) (V −O)− EC − 0.18\n(V −O)0\n\nDivacancy Oxygen complex (V2 −O)\nVacancy Oxygen dimer complex (V −O2i) (V −O2i)0\n\nThe modeling of the complex chemistry of defects in irradiated silicon can be quite\n\ncomplex and unpractical for the modeling of irradiated pixel sensors. However simple pa-\n\nrametrization reproducing well the different known effects observed in silicon has been\n\ndevelopped in the recent years. A deep acceptor and a deep donor are introduced in the\n\nbulk to account for space charge sign inversion (SCSI), double junction effects [5; 6] and\n\nleakage current increase [5; 6; 30; 31]. In addition, to account for trapping and recombi-\n\nnation of carrier induced by radiation damage, a shallow hole and electron trap must be\n\nadded. The model we used is shown in tables 2.4 and 2.5 , based on the latest results from\n\nRD50 collaboration [29] and work of several groups [32; 33; 34].\n\n64\n\n\n\n2.5. RADIATION DAMAGE IN SILICON SENSORS\n\nTable 2.4 – n-type radiation damage model\nDefect’s energy\n(eV)\n\nIntroduction rate\n(cm−1)\n\nElectron capture\ncross-section (cm−2)\n\nHole capture cross-\nsection (cm−2)\n\nEc − 0.42 13 2.2e-15 1.2e-14\nEc − 0.53 0.08 4e-15 3.5e-14\nEc − 0.18 100 1e-14 1e-16\nEv + 0.36 1.1 2e-18 2.5e-15\n\nTable 2.5 – p-type radiation damage model\nDefect’s energy\n(eV)\n\nIntroduction rate\n(cm−1)\n\nElectron capture\ncross-section (cm−2)\n\nHole capture cross-\nsection (cm−2)\n\nEc − 0.42 1.613 2.e-15 2e-14\nEc − 0.46 0.9 5e-15 5e-14\nEc − 0.10 100 2e-15 2.5e-15\nEv + 0.36 0.9 2.5e-14 2.5e-15\n\nThese parameters must be adjusted using a fit method to determine adequate intro-\n\nduction rates, level and capture cross-section for each type of silicon used.\n\nA simpler parametric model has been developed by the ROSE and RD50 collaboration\n\nto evaluate the evolution of leakage current, average trap characteristic lifetime and net\n\ndopant/defect concentration in irradiated sensors. The Variation of leakage current density\n\nin the bulk of a depleted sensor is parametrized following equation 2.33\n\n∆Ivol\nV\n\n= αΦ (2.33)\n\nWhere Ivol is the volume generated current, V the bias potential, α the leakage damage\n\nconstant and Φ the exposed fluence in neq/cm\n2. The average trap characteristic lifetime\n\ncan be also be expressed as a function of fluence as shown in equation 2.34, where τt0 is\n\nthe original trapping time.\n\n1\n\nτt(Φ)\n=\n\n1\n\nτt0\n+ βΦ (2.34)\n\nFinally, the net dopant/defect concentration is expressed following equation 2.35, where\n\nND0 is the initial donor concentration,NA0 the initial acceptor concentration andN the net\n\ndopant concentration. From N, it is possible to compute the depletion potential following\n\n65\n\n\n\n2.5. RADIATION DAMAGE IN SILICON SENSORS\n\nequation 2.5. Acceptor removal and donor creation have not been observed experimentally,\n\nexplaining the absence of such term in this equation. 0\n\nN(Φ) = ND0e\n−cΦ −NA0 − bΦ (2.35)\n\nTable 2.6 show typical value for the radiation damage constants of this model.\n\nTable 2.6 – Typical radiation damage constants [35]\n\nconstant value\nα 8.0× 10−17 Acm\n\nβ 0,24 ×10−6 cm2s−1\n\nc 3.54 ×10−13 cm2\n\nb 7.94 ×10−2 cm−1\n\n2.5.2 Ionizing energy loss\n\nSilicon dioxide present at the surface is the main material damaged by ionizing energy\n\nloss . The dose of ionizing energy loss radiation damage is usually expressed in Rad, which\n\nrepresent 6.24× 1010 MeV of ionizing energy deposition per kilogram of material.\n\nSiO2’s Oxygen valence electrons present at the interface and uncompensated by a Si-\n\nlicon atom create local traps for holes, as shown in figure 2.13. Holes from electron-hole\n\npairs generated by ionizing particles crossing the oxide can be trapped in this layer. Elec-\n\ntrons have higher mobility (20 cm2/V s) than holes (2× 10−5 cm2/V s) in SiO2 and collect\n\nrapidly while holes accumulates in the traps present near the interface [36]. The electrical\n\nfield created by this sheet of positive charge attracts silicon’s free electrons that then for\n\na compensating layer of free carrier at the Si-SiO2 interface. Few carriers can cross the\n\ninterface potential barrier by tunneling and recombine with their opposite carrier, leaving\n\nthe charge layer, called inversion layer, almost permanently at the interface.\n\nThe charge density at the interface is known to vary almost linearly with exposed\n\nfluence from 1011 cm−2 to 1012 cm−2 between 0 and 1 × 108 Rad, usually equivalent\n\nin ATLAS inner detector to exposition to a NIEL of 0 to 1 × 1015 neq/cm\n2. We then\n\nconsider the charge layer to be saturated for higher fluences, as observed experimentally\n\n66\n\n\n\n2.5. RADIATION DAMAGE IN SILICON SENSORS\n\n[37]. Oxide charge saturation concentration is process dependent [38] so the saturation\n\nfluence and charge concentration can be chosen for modelisation to be coherent with the\n\ndata presented in literature.\n\nFigure 2.13 – Schematics of radiation damage effect at the silicon-silicon dioxide interface\nin silicon sensors.\n\nThe presence of this electron layer at the interface can form a conductive path between\n\ndifferent electrodes, increasing crosstalk and leading to unwanted parasitic leakage path in\n\nthe sensor. Mitigation methods to prevent this effect will be presented in next chapter.\n\n67\n\n\n\n2.5. RADIATION DAMAGE IN SILICON SENSORS\n\n68\n\n\n\nChapitre 3\n\nTCAD Simulation models\n\nTechnology Computer-Assisted Design (TCAD) uses our present knowledge of the par-\n\ntial differential equations describing charge carrier’s motion and interactions with the crys-\n\ntal lattice in semiconductors, detailed in equations 2.2, 2.3 and 2.4 coupled to finite element\n\nmethod to simulate the electrical parameters of the device. Finite element method use a\n\nlinearized version of the transport equation to describe the problem in terms of a linear\n\nsystem of equation that can be solved by linear algebra methods. To obtain a solution to\n\nthe variables of the transport equations (n,p,V) in a arbitrary geometry, we must subdivide\n\nthe surface or volume in rectangular, triangular, prismatic or pyramidal sub-elements small\n\nenough that the solution in locally polynomial in this domain and can be approximated by\n\na polynomial Φ. The sum of all sub-elements covering the simulation geometry is call the\n\nmesh, as seen in the example for a simple geometry in figure 3.1.\n\nTo obtain a good approximation of the solution using this method, mesh element’s\n\ndomain size must be chosen to be sufficiently small to be able to do this approximation.\n\nRegion where the solution is expected to vary rapidly must be subdivided in small region\n\nuntil the solution can be represented as a locally polynomial function. TCAD software\n\nare bundled with meshing algorithm that can use know quantities in the geometry, such\n\nas impurity concentration, to generate the sub-elements covering the domain to simulate.\n\nHowever, no perfect method exist to determine the perfect mesh and case-by-case study of\n\nthe mesh to use must be perform to ensure the validity of the solution obtained over this\n\ndiscretization. The solution to the equations, once meshing and interpolation equations are\n\n69\n\n\n\nFigure 3.1 – Meshing of a disc surface using triangular sub-elements\n\nchosen, can then be expressed as :\n\nV, p, n =\n\nn∑\ni\n\naV,p,ni Φi (3.1)\n\nWhere i is the indice of an intersection of the sub-elements. The function Φ are usually\n\nchosen to be equal to 1 at element intersection i and 0 at all other surrounding intersection.\n\nFigure 3.2 show an example of functions Φ that can be selected in a simple 1D geometry\n\nto approximate a semi-spherical function.\n\nThis method can be used to explore different designs of semiconductor detectors before\n\ntheir production and optimize its electrical parameters. We used this method to explore\n\ndifferent possible designs for the IBL and super-LHC ATLAS planar pixel detector. The\n\ninfluence of many design parameters as the number and spacing of guard rings, sensor’s\n\nthickness and inactive edge width on sensor electrical characteristics have been simulated,\n\ngiving insight on the effects of sensor design and processing on the devices performances.\n\nRadiation damage can also be included in the physical model used in the simulation,\n\nallowing to extract macroscopic effects caused by change in bandgap structure detailed in\n\n70\n\n\n\n3.1. PROCESS SIMULATION\n\nFigure 3.2 – Simple linear interpolation function used to approximate a function in 1D\n\nchapter 2.\n\n3.1 Process simulation\n\nThe first step to obtain a realistic simulation of a sensor is to obtain an accurate\n\ndescription of its geometry. Most TCAD simulation software include a process simulation\n\npackage that allow to simulate the fabrication process of silicon sensors. The main step to\n\nproduce a realistic process simulation of a pn junction are as follow :\n\n1. Oxidation\n\n2. Nitride deposition and etching\n\n3. Insulation implantation\n\n4. Oxide etching\n\n5. Implantation\n\n6. Thermal annealing\n\n7. Via etching and electrode deposition\n\n8. Passivation and passivation etching\n\nIn typical silicon detectors, an additional implant must be created to insulate the elec-\n\ntrodes from each other before creating the main implant. As it was seen in chapter 2,\n\n71\n\n\n\n3.1. PROCESS SIMULATION\n\nionizing radiation damage cause the formation of an inversion layer at the silicon-oxide in-\n\nterface that can shortcut the different implants. To prevent this, an low dose p+ implant is\n\ncreated on the n+ side of the pixel sensors. This implant’s excess of free holes will compen-\n\nsate the inversion layer created by the accumulation of electrons and interrupt the channel\n\nthat forms at the surface. Three type of insulating implants are used in the industry. The\n\nmost simple is the p-spray technique, which consist in the implantation of a uniform p\n\ndose across the full wafer. A different technique , called p-stop, use an additional nitride\n\nlayer deposited over the oxide layer to stop the dopant ion beam and create segmented p+\n\nimplants where nitride was etched between the main n+ ones. Finally, an hybrid method,\n\ncalled moderated p-spray, combines both method. The thickness of the layer of nitride is\n\nchosen to let some of the ions reach the silicon, creating a uniform implant across the wa-\n\nfer and stronger and deeper implants where nitride was etched, between the n+ implants.\n\nFigure 3.3 shows the two first type of insulation. The moderated p-spray is simply a combi-\n\nnation of both method. The insulation implants form junctions with the main n+ implant\n\nand high electric field will be present in this region. Each method need to be optimized\n\nto avoid creating high electric field that may lead to breakdown while providing sufficient\n\ninsulation for operation after irradiation.\n\nThe second step in the process consist in heating the bare wafer of silicon in a oxygen\n\natmosphere to grow a layer of silicon dioxide at the surface of the wafer (O(200 nm))\n\n. This layer is then etched down to a very thin layer (O(40 nm)) using lithography, to\n\nform the shape of the implants. A beam of mono-energetic ions (O(10− 120 keV )) is then\n\ndirected to the surface of the wafer and a fixed dose of ion by surface unit (O(1015 cm−2))\n\nis delivered. In the area where a thick film of oxide has been left, the stopping power of the\n\nlayer is sufficient to completely absorb the incoming ions while in the etched region, ions\n\nwill travel into silicon forming an implant with a shape and magnitude determined by the\n\nimplantation dose, the beam energy and orientation and the thickness of the oxide layer\n\nleft in the implant area.\n\nOne all implants have been created, the wafer must undergo an annealing step, where it\n\nis heated for sometime at very high temperature. This step is needed to activate the dopant\n\nintroduced in silicon. To be electrically active, dopant atoms must be correctly placed in\n\n72\n\n\n\n3.1. PROCESS SIMULATION\n\n(a) p-spray insulation\n\n(b) p-stop insulation\n\nFigure 3.3 – The two main type of implant insulation used in pixel sensors.\n\nthe crystal lattice. The heating process gives the kinetic energy to the dopants to diffuse\n\nthrough the bulk and react with the lattice structure to become electrically active. During\n\nthis process, the shape of the implant is modified as dopants migrate by diffusion further\n\ninto the silicon bulk. The annealing temperature and time are important parameters to\n\ndetermine the final shape of the junction.\n\nThe final step of the process consist in opening small holes, called via, in the oxide in\n\nthe n+ implant area reaching the surface of the silicon (nitride has previously been opened\n\nin this area to allow access to the oxide at this step). An aluminum layer (O(800 nm))\n\nis then deposited and etched outside electrode area to produce the electrical contacts to\n\nthe n+ implants. Finally, a thick layer of a passivating material is deposited over the area\n\nuncovered by the electrodes to provide a good protection of the surface.\n\n73\n\n\n\n3.2. DEVICE SIMULATION\n\nThe parameters of the process affecting the implant profile need to be known to create\n\naccurate representation of the device we wish to simulate. The process details we use in\n\nour simulation have been obtained through discussion with designers and manufacturer\n\nof silicon devices. Some parameters are however hard to determine from accessible data\n\nand are not disclosed by the manufacturer. These values can however be obtained through\n\nexperimental methods as will be shown in the rest of this chapter.\n\n3.2 Device simulation\n\nDevice simulation is used to obtain electrical parameters of a geometry we built through\n\nprocess simulation. For a device simulation, the geometry to be simulated must be carefully\n\nchosen to avoid increasing the computational complexity of the problem to be solved.\n\nBoundary conditions must also be selected to represent the operation conditions of the\n\ndevice.\n\n3.2.1 Geometry\n\nFull simulation of a pixel sensor is impossible to perform in modern computers as\n\nthe number of mesh point needed to accurately describe the solution to the equations in\n\nthe three-dimensional domain is too important to be handled by computers. Simulation\n\nperformed in the next section are done in two dimension to reduce the simulation time.\n\nPeriodicity and symmetry of the geometry can be exploited to reduce the size of the problem\n\nto be solved. Figure 3.4 shows how pixel sensor quasi-periodicity and symmetry can be used\n\nto define a two-dimensional geometry that can be simulated with a TCAD software. We\n\nconsider a Y-Z oriented cut plane in a semi-infinite sensor with its guard ring structure\n\nand cutting edge and a plane of pixels extending extending infinitely in the ±X and +Y\n\ndirection. This represent a good approximation for a pixel sensor far from the corners of\n\nthe device. The edges not represented in the two dimensional representation are supposed\n\nto not interfere with the local electrical behavior of the simulated geometry.\n\nTo obtain the solution to the differential equations describing the charge transport and\n\nPoisson equation, we must provide the boundary conditions as fixed values of the variables\n\n74\n\n\n\n3.2. DEVICE SIMULATION\n\nto be solved, the electron and hole concentration (n,p) and the electrostatic potential(V)\n\nor their derivative, current densities ( ~Jp,n) and electric field ( ~E). Real operation conditions\n\nof the simulated sensors cannot be completely described in this manner and approximation\n\nmust done to obtain a solvable problem. The simulation geometry must be selected to allow\n\na solution that represent correctly the real operation conditions.\n\nFigure 3.4 – Simplification of a n-in-n planar pixel sensor geometry for TCAD simulation\n\n3.2.2 boundary conditions\n\nTo solve our set of differential equations we need to restrict ourselves to a solution\n\nin a bounded domain, the sensor. We must choose boundary conditions reflecting the\n\nproperties of the system we want to simulate. Three types of boundaries were used during\n\nour simulation, representing the oxide-silicon interface, the electrode interface, and the\n\nperiodicity boundary. In addition we need a model for the cutting edge of the sensor.\n\nThe boundaries between silicon dioxide and silicon is a semiconductor/insulator boun-\n\ndary characterized by the presence of an accumulated charge layer at the interface. The\n\n75\n\n\n\n3.2. DEVICE SIMULATION\n\nboundary condition applied to these surfaces for the Poisson equation is the Neumann\n\nboundary condition (3.2) that takes into account the charge layer density (ρs) present at\n\nthe surface . Also, electrons and holes concentrations are set to zero on this boundary and\n\nthe current is not allowed to flow through this surface.\n\nn̂ · ε1~∇Φ1 − n̂ · ε2~∇Φ2 = ρs (3.2)\n\nMetal-semiconductor surfaces are the boundaries between the silicon bulk and the me-\n\ntallic electrodes. This is usually a ohmic contact and the current is allowed to flow through\n\nthem. The voltage Φ is constant and equals the bias voltage applied to the sensor by an\n\nexternal power supply. The concentration of carriers (ps,ns) at the surface of the contact is\n\ndetermined by equations (3.3), (3.4), derived for Boltzmann’s statistics, knowing the bias\n\nvoltage applied at the electrodes. The effect of the contact work function is considered\n\nnegligible as highly doped regions are located below the electrodes.\n\nns =\n1\n\n2\n[(N+\n\nD −N\n−\nA ) +\n\n√\n(N+\n\nD −N\n−\nA )2 + 4n2\n\ni ] (3.3)\n\nps =\nn2\ni\n\nns\n(3.4)\n\nWhere N+\nD , N\n\n−\nA are the ionized donors concentration and ionized acceptors concentra-\n\ntion in cm−3.\n\nGuard ring structures are metal semiconductor interfaces where the metallic electrode\n\nself-biased. To represent this case, we must impose a null current flow on this contact. The\n\nbias voltages taken by the floating contacts are then found by the solver of the TCAD\n\nsoftware.\n\nTo reduce the size of the problem to be solved, we can use periodicity boundary condi-\n\ntions using geometric properties of the sensor. In our simulation, we will be interested to\n\nthe solution on the sides of the sensor. Knowing the solution will become quasi-periodic\n\nin the X-Y plane when approaching the center of the device. Far from the edge , we can\n\ntruncate our model at a distance large enough to consider the solution will become as if it\n\n76\n\n\n\n3.2. DEVICE SIMULATION\n\nwas periodic at this point. We then impose the periodicity condition (3.5) at the surface\n\nfor electrons and holes concentration and for the bias voltage.\n\n~∇V · n̂ = 0\n~∇n · n̂ = 0\n~∇p · n̂ = 0\n\n(3.5)\n\nWhere n̂ is the unitary normal vector of the boundary. Physically, this represent the\n\ncondition where no current is flowing out or in the geometry and no electric field lines flow\n\nout of the simulated boundary.\n\nThe dicing of pixel sensors from their originating wafer creates structural damage that\n\naffects the properties of the edge. A dead edge width must be included in the design to\n\nexclude this zone from the sensible part of the sensor. This dead edge is added to inactive\n\npart of the sensor and must be kept as small as possible.\n\nA special attention must be taken to model the cutting edge of a silicon sensor. Dicing\n\nmechanism induces structural damages in the Silicon crystal lattice near the cutting region.\n\nThis induces a process of amorphization of silicon. Amorphous silicon is a complex material\n\nwhere no short or long distance orders exists in the crystal lattice. A method to model\n\namorphous silicon is to introduce a high number of defects in the band gap of Silicon. As\n\nthe crystal lattice of the Silicon is highly perturbed in the cutting edge region, trap states\n\nare created by the defects in the crystal lattice that are introduced. To represent such a\n\ndistribution of defects in the band gap, we use a continuous density of states distribution\n\nto describe the band gap defects distribution. This distribution can then be tuned to reflect\n\nthe behavior of real sensors measured in the laboratory. The generation-recombination term\n\nrelated is calculated using an integral form of equation 2.10. Equation (3.6)[39] shows how\n\nwe describe the defect distribution in the band gap.\n\ng(E) = gTA(E) + gTD(E) + gGA(E) + gGD(E)\n\ntA(E) = NTAe\nE−Ec\nWTA\n\ntD(E) = NTDe\nEv−E\nWTD\n\ngA(E) = NGAe\n(\nEGA−E\nWGA\n\n)2\n\ngD(E) = NGDe\n(\nE−EGD\nWGD\n\n)2\n\n(3.6)\n\n77\n\n\n\n3.2. DEVICE SIMULATION\n\nThe density distribution function consists of two exponential tails functions (TD, TA)\n\nand two Gaussian function distributions for donors and acceptors (GD,GA) giving the\n\nenergy distribution in cm−3. Table 3.1 shows the default parameters used for this model\n\nin our simulation. The defect density distribution that is created by these parameters is\n\nrepresented in figure 3.5. Ev = −1.12 eV is the valence band energy and Ec = 0 the\n\nconduction band energy. The model used in these simulations was proposed by E. Noschis\n\nand al. [40]\n\nTable 3.1 – Default defect density of states distribution parameters in SILVACO TCAD\nsoftware\n\nParameters Values\nNTA 1.12× 1021 cm−3/eV\nNTD 4.00x1020 cm−3/eV\nNGA 5.00× 1017 cm−3/eV\nNGD 1.50× 1018 cm−3/eV\nEGA 0.4 eV\nEGD 0.4 eV\nWTA 0.025 eV\nWTD 0.050 eV\nWGA 0.100 eV\nWGD 0.100 eV\n\nFigure 3.5 – Defect density distribution in the band gap of amorphous silicon used for\nour simulation\n\n78\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\n3.3 The Multi-Guard Ring structure\n\nThe goal of the guard ring structure present next to the high-voltage electrode or the\n\npixel matrix is to ensure a smooth transition from high voltage to ground while approaching\n\nthe outer edge of the device. The edge usually takes the same potential as the backside\n\nof the sensor as the high amount of defects at the cutting edge render it conductive and\n\nelectrically link the two sides together. Each guard ring acquire its bias voltage by a punch-\n\ntrough mechanism forming a smooth transition from high bias voltage to ground at the\n\nedge of the sensor. This is needed to ensure that no bias voltage difference exist between\n\nthe two sides of the wafer, close to the edge. This bias would generate excessive current\n\ndetrimental to the operation of the sensor [36]. The inner ring can also be used, if connected\n\nto bias, to collect leakage and surface current that would increase the noise when collected\n\nby the pixels. This principle is called a current-terminating structure (CTS) [40].\n\nIn ATLAS actual design of the pixel sensor, the number of guard ring is fixed to 16, with\n\nan implant width of 10 microns. The electrodes covering the guard ring implants are 16 to\n\n22 µm wide with the largest at the outer side of the structure. They overshoot the implant\n\nby 2 µm in the edge direction and 1 to 40 µm in the active area direction. Distance between\n\nthem varying from 15 to 8 µm. The guard rings represent a dead zone in pixel sensors,\n\nmeaning no particle is detected close to the structure. This corner of a IBL prototype\n\nsensor with its guard ring is represented in figure 3.6. The presence of these inactive zones\n\nsurrounding the sensor force the overlapping of the sensors in ATLAS tracker to avoid\n\ndetection gap between sensors. This overlap increase the amount of material present in\n\nthe tracker and should be avoided to reduce the material budget of the inner detector and\n\nincrease the simplicity of its configuration. Reduction of the guard ring area for the sensors\n\nto be used in IBL are a key to minimize its material budget and its inactive zones.\n\nThe goals of the simulation performed in this chapter are to evaluate the effects of\n\nmodifying the number of guard rings and their spacing factors that can reduce the dead\n\nzone while maintaining adequate operation conditions for the sensor and exploring the effect\n\nof radiation damage on the efficiency of guard ring structure. This was used to determine\n\nhow modifications to the sensor and guard rings geometry can be used to reduce inactive\n\n79\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\narea while keeping guard rings active.\n\n80\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\n3.3.1 Principles of guard ring structures\n\nFigure 3.6 – Corner view of a FE-I4 n-in-p pixel sensor\nstructure showing the guard ring structure and the 3 first\ncolumn and 18 first row of pixels.\n\nThe guard rings are used to\n\ncontrol the potential drop from\n\nelectrodes in the active area of\n\nthe sensor to the cutting edge\n\nof the device, as shown in fi-\n\ngure 3.7 for the example of a\n\nn-in-n sensor. This provide a\n\nreliable way to control the po-\n\ntential at the surface of the\n\nsensor edges. Without guard\n\nrings, the potential distribu-\n\ntion at this surface would be\n\nlargely influenced by its pro-\n\ncessing quality. The presence\n\nof an inversion channel,which\n\nmagnitude is related to surface quality, render the surface conductive, and potential drop\n\nonly occur in resistive regions of the surface. The presence of defects at the interface will\n\naffect the resistivity of the surface and can be detrimental to detector operation by cau-\n\nsing the presence of sharp electric field peaks in the more resistive regions and eventually\n\nbreakdown of the sensor. The guard rings help to control the surface behavior and render\n\nthe sensor detector more stable and independent of the surface state by imposing a gradual\n\nsurface potential drop through a bulk process, independent of the surface state.\n\nFig. 3.8 shows guard ring geometry, with its metal overhangs covering the oxide. Guard\n\nrings are biased by a punch-trough mechanism creating a current circulating between the\n\nguard rings. The punch-trough occurs when the depletion region of both guard rings are\n\nin contact. This happens as the depletion region of the pixels or high voltage electrode\n\nexpands as bias increase and reach the different guard rings. The vicinity of a guard ring\n\nmetal electrode is biased at the same potential as the silicon implant region, reducing the\n\nvertical electric field in the oxide under the guard ring overhangs. This has the effect of\n\n81\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\nFigure 3.7 – Guard ring electrical behavior (n-bulk)\n\ninterrupting the inversion channel formation at the Si − SiO2 interface, problematic for\n\nn- in-p sensors, as shown on figure 3.8. This channel is caused by the presence of hole\n\ntraps near the SiO2 boundary with Silicon, as explained in chapter 2. These traps become\n\npositively charged as they are filled by holes created by an incident ionizing particle or\n\nthermal generation [41]. In the case of n-in-p sensors, the guard rings are located on the\n\nsame surface as the pixels, collecting electrons. The orientation of the electric field inside\n\nthe oxide will favor the drift of holes to the silicon silicon dioxide interface and increase the\n\nmagnitude of the inversion layer charge density depending on the local current density in\n\nthe oxide. Close to the guard rings, reduced electric field will inhibit the drift of the hole\n\nin the oxide and cause this interruption of the inversion channel\n\nA set of two guard rings can be seen as a blocked MOSFET with the source connected to\n\ngate. The transistor is kept in an off state with the punch-trough voltage and gate to drain\n\nresistance determining the guard ring bias behavior. The guard ring metallic overhangs, are\n\nused to reduce the electric field present on this side of the guard ring at the interface and\n\ninterrupt the electron channel by suppressing the vertical electric field in this region. The\n\nlong overhang, oriented towards the pixel region, is used to suppress the punch-trough hole\n\ncurrent and increase the punch-trough voltage [36]. The p-spray, p-stop and moderated\n\np-spray methods are also used to mitigate the effects of the formation of this inversion\n\nchannel. A drawback of this method is the apparition of electric field peaks at the sides\n\n82\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\nof the guard ring’s implant junction with the p-spray implant, making this region a weak\n\npoint for breakdown formation.\n\nFigure 3.8 – Guard ring schematic representing the interruption of the inversion channel\n(p-bulk)\n\nThe multi guard ring structure geometrical and electrical parameters such as the doping\n\nprofile of the implant, the overhangs length and the distance between guard rings influence\n\nthe electrical behavior of the guard rings. The voltage drop between guard rings will be\n\nmodified if these parameters are changed as punch-through voltage and currents will be\n\naffected by these modifications. The punch-trough mechanism occurs at depletion and\n\nguard rings become active when lateral depletion zone reach them. As we know from\n\nequation 2.5, depletion depth is proportional to the square root of the applied bias voltage.\n\nFor guard rings located close to the pixel region, the overhang directed toward the pixel\n\nmust be kept small to favor higher punch-trough currents and bigger voltage drop on shorter\n\ndistance. TCAD simulation represent the perfect tool to study the potential distribution\n\nof guard ring geometries and to optimize them to improve the performances of the pixel\n\nsensors\n\n83\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\n3.3.2 Optimization of guard ring structures for reduction of inactive\narea and radiation hardness\n\nTCAD simulation focuses on comparing electrical parameters of different multi-guard\n\nring structures before and after irradiation. Figure 3.9 shows the geometry of the three\n\nstructures simulated. The first one is the ATLAS actual pixel sensor, an n-in-n structure.\n\nThe two others are n-in-p designs proposed as candidate for IBL and super LHC pixel\n\nsensor replacement. Detectors with these new multi-guard ring structure designs have been\n\nproduced in collaboration with the laboratories forming the ATLAS Planar Pixel Sensor\n\nUpgrade group. P-bulk and n-bulk productions of pixel sensors including other structures,\n\nlabeled in this document as respectively PPSU09-n and PPSU09-p, were organized and\n\nbased on the knowledge gathered from the simulation presented here and experimental\n\nstudies from collaborating laboratories.\n\nSimple 6x6 mm diodes with a number of guard rings varying from 1 to 4 were placed on\n\nthe production to help with simulation model calibration by providing a simple geometry\n\nfor comparison with TCAD results. The large size of the diode guard ring allow easy\n\nmeasurement in clean room of their potential and were also used to study the evolution of\n\nguard ring behavior for this production under irradiation.\n\nThe simulation studies performed prior to the production of the sensors [42; 43; 44;\n\n45; 46] have shown that the results of the simulation, regarding guard ring behavior and\n\nbreakdown voltage was dependent on doping profile of the implants forming the guard ring\n\nstructure and pixels. A dedicated test structure was placed on each of the PPSU09 wafer\n\nproduction to eventually measure the profile of the implants using various techniques. Re-\n\nsults of these measurements will be presented in the next chapter. Simulation presented\n\nin this section are based on the measured implant parameters for the PPSU09 production\n\nand simulation done prior to it have been reprocessed with the correct implantation para-\n\nmeters. Bulk resistivity was fixed to 5000 Ωcm for all simulation. Temperature was 300 K\n\nfor unirradiated sensors and 250 K for irradiated ones.\n\nIn this section, I present simulation performed to reduce the span of the multi guard-\n\nring structure through reduction of the number of guard rings and shift of the structure\n\nunder the pixels. Radiation damage effects on guard rings were simulated to evaluate the\n\n84\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\nbehavior of the new guard ring structure under irradiation. Following the results of these\n\nsimulations, the method was used for the final n-in-n planar sensor candidate for the IBL\n\nin the PPSU09 production.\n\nFigure 3.9 – Multi-guard ring structures used for simulation, obtained from process si-\nmulation\n\nSimulation of the current ATLAS planar pixel sensor design was first performed up to\n\n500 V for unirradiated and irradiated sensors up to 1015 neqcm\n−2, the foreseen fluence for\n\nthe actual sensor in LHC. The radiation damage model presented in table 2.4 was used.\n\nFigure 3.11 shows the simulated bias voltage for the Actual ATLAS pixel design for irradia-\n\nted and unirradiated case. The irradiated case is simulated at the working operation bias\n\nvoltage of 150 V. The irradiated model simulation were performed at 500V, the maximum\n\n85\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\npossible bias voltage reachable with the current power supplies. One pixel is included along\n\nwith the guard rings (to the left of the figures). As fluence seen by the sensor increase, the\n\nspace charge sign invert and depletion occurs from pixel toward the backplane like in a\n\nn-in-p sensor design. Figure 3.10 show the potential distribution on the guard rings of the\n\nATLAS model as a function of the bias voltage. Comparison with experimental data is also\n\nshown. More details on experimental validation will be given in next chapter.\n\nFigure 3.10 – Potential distribution in the ATLAS standard n-in-n Multi-Guard Ring\nStructure, simulated and measured\n\nFigure 3.12 shows the electron concentration in the sensor for different fluences. The\n\nresults of Space charge sign inversion (SCSI) is the replacement of electrons by holes as the\n\nmajority carrier. This is shown in figure 3.13 representing hole concentration increasing as\n\nelectron concentration decrease, mainly in the undepleted region. The undepleted volume\n\nfor an unirradiated sensor extends on 900 microns from the edge of the sensor. The depletion\n\nzone never reach the edge of the device because of the large 500 µm safety edge left after\n\nthe guard rings. This edge width could be modified to reduce the inactive edge of the sensor\n\nwhile keeping a safe margin between the cutting edge and the depletion region. This width\n\nis retained until space charge sign inversion but it is then replaced by a holes undepleted\n\narea , as seen in figure 3.13.\n\nThe n-in-p structures shown in figure 3.9 were also simulated in the IBL conditions\n\n86\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\n(a) unirradiated (150 V) (b) 1014neqcm\n−2 (600V)\n\n(c) 5× 1014neqcm\n−2 (600V) (d) 1015neqcm\n\n−2 (600V)\n\n(e) 5× 1015neqcm\n−2 (600V)\n\nFigure 3.11 – Simulated 2D voltage profile for ATLAS n-in-n pixel sensor (300 µm thick-\nness, 1700µm width. Color scale in Volt)\n\n87\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\n(a) unirradiated (150 V) (b) 1014neqcm\n−2 (600V)\n\n(c) 5× 1014neqcm\n−2 (600V) (d) 1015neqcm\n\n−2 (600V)\n\n(e) 5× 1015neqcm\n−2 (600V)\n\nFigure 3.12 – Simulated 2D electron concentration profile for ATLAS pixel sensor (300\nµm thickness, 2500µm width)\n\n88\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\n(a) unirradiated (150 V) (b) 1014neqcm\n−2 (600V)\n\n(c) 5× 1014neqcm\n−2 (600V) (d) 1015neqcm\n\n−2 (600V)\n\n(e) 5× 1015neqcm\n−2 (600V)\n\nFigure 3.13 – Simulated 2D hole concentration profile for ATLAS pixel sensor (300 µm\nthickness, 2500µm width)\n\n89\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\n(250 microns edges, 1000V bias voltage, up to 1016 neqcm\n−2). These n-in-p guard ring\n\ndesign were proposed to reduce the span of the guard ring structure currently used by\n\nreducing the size of the different guard rings or by reducing their number of them. The\n\nfirst design uses 17 very small guard rings while the second model use 9 larger guard\n\nrings and both represent an approximative reduction of the guard ring span by 200 µm.\n\nFigure 3.14 shows the hole concentration in the small guard ring sensor at various levels\n\nof irradiation. Resistivity of the silicon is reduced by the radiation damage and depletion\n\npotential should be increased. This result in a reduction of the visible depth of the depleted\n\nregion for a given bias voltage. The depletion region never reaches the cutting edge of the\n\nsensor indicating that the 100 µm safety margin is safe enough for this type of n-in-p\n\nsensors. The large guard ring model show hole concentration distribution compatible with\n\nthe small guard ring model and is not represented here.\n\nFigure 3.16 show the distribution of potential on the guard rings of the three structures.\n\nThe unirradiated structures are biased to voltage higher than breakdown voltage to observe\n\nthe guard ring behavior and put in evidence the bend in the curves due to breakdown that\n\nshould be observable experimentally. This simulation is possible because TCAD simulation\n\ndoes not include thermal simulation. Voltage curves shown in figure 3.15 are cut before\n\nthe 1500 V bias voltage because of the divergence when simulating bias voltage much over\n\nbreakdown voltage. This is due to a hard breakdown occurring on edge pixel side generating\n\nhuge current. Guard rings are biased as the depletion zone limit approach toward them\n\nand as current begins to circulate between them by punch-trough mechanism. In n-in-n\n\nsensors, as depletion occurs from pixel side after SCSI, guard rings on the opposite side\n\nstay in the undepleted zone until full depletion of the sensor, making them ineffective to\n\nprovide the smooth surface potential drop required for operating sensors with a high full\n\ndepletion voltage. This situation also implies that a steep voltage variation is present at the\n\nsurface between the pixels (0 V) and edges (High Voltage). This eventually creates a weak\n\nspot favoring breakdown in the sensor, as observed in the Current versus Voltage curves\n\nin figure 3.15. The guard rings of the n-in-p structures have a similar behavior before\n\nand after irradiation as the depletion still occurs from the pixel side towards the high\n\nvoltage electrode side. The difference in the bias potential of each guard ring at different\n\n90\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\n(a) unirradiated (1000 V) (b) 2× 1015neqcm\n−2 (1000V)\n\n(c) 4× 1015neqcm\n−2 (1000V) (d) 6× 1015neqcm\n\n−2 (1000V)\n\n(e) 8× 1015neqcm\n−2 (1000V) (f) 1016neqcm\n\n−2 (1000V)\n\nFigure 3.14 – Simulated 2D hole concentration profile for n-in-p small guard ring type\npixel sensor (300 µm thickness, 2500µm width)\n\n91\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\nfluences is due to the damage in the oxide which accumulate charge at the silicon-oxide\n\ninterface and to a modification of the position of the depletion region limits. The change\n\nin space charge density modify the effective doping concentration that determines the full\n\ndepletion potential of the sensor. As guard rings are biased by a punch-trough mechanism,\n\nthe position of the depleted region limit relative to the guard rings position determines the\n\nbias potential of guard rings as a function of high voltage bias. The bent observed in guard\n\nring potential curves at high voltage is directly related to the current being generated in\n\nthe guard ring vicinity by impact ionization and can be correlated to the breakdown in\n\nfigure 3.15.\n\nFigure 3.17 show the electric field distribution at 1 µm under the guard ring for the\n\nthree model under study at 400V. The electric field peak value are the lowest for the ATLAS\n\nactual design however the structure is also the largest. The two n-in-p model show different\n\ndistribution of peaks with the highest one located in the small guard ring structure. The\n\nlarge guard ring structure exhibit more peaks with higher value.\n\nSimulations also have shown that depletion for highly irradiated sensors is an ill de-\n\nfined concept as carrier concentration becomes very low in what is considered to be the\n\nundepleted zone, as seen on figure 3.13 and 3.14. Some low electric field (O(1000 kV/cm))\n\nis present in the simulation in this region that is usually considered undepleted and charge\n\ndeposited in this region could still be collected as its recombination probability is much lo-\n\nwer than in the undepleted regions of an unirradiated sensor. Figure 3.18 show the electric\n\nfield distribution as a function of the depth under a pixel for the ATLAS standard model\n\nirradiated at a fluence of 5× 1015neqcm\n−2. The electric field in the undepleted portion of\n\nthe bulk, from 100 to 300 µm is only an order of magnitude lower than in the undepleted\n\nregion. Since the velocity of carrier saturate and is reduced by an order of magnitude in\n\nthe high electric field present in the depleted region, the drift in each zone would be almost\n\nequivalent.\n\n92\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\nFigure 3.18 – Electric field magnitude in the bulk of an ATLAS sensor, under a pixel,\nirradiated to a fluence of 5 × 1015neqcm\n\n−2 and mobility as a function of parallel electric\nfield for electrons\n\nThe simulation of the guard ring structure after irradiation suggests that n-in-p struc-\n\ntures should offer better resistance to irradiation at high fluences as the guard ring structure\n\nwill continue to be effective during all the detector’s operation time. This also suggests that\n\nplacing a guard ring structure on the n side of the n-in-n pixel design could allow operation\n\nafter exposition to the required dose. The small guard ring design endure an higher break-\n\ndown voltage when compared to the large guard rings design. Behavior after irradiation\n\nis however very similar. Smaller guard rings exhibit lower electric field peak in average,\n\nbut the structure simulated is ideal. In reality, smaller guard rings might have higher pro-\n\nbability of defects in process, thus creating weak spot where breakdown could occur. The\n\nhighest peak was also found in the small guard ring model.\n\nTu be used in IBL conditions, the ATLAS standard structure dead region must be\n\nreduced to the level of the n-in-p structures. Simulation have been performed to explore\n\nthe possibility of reducing the dead area of the sensor that span from the edge to the first\n\nguard ring. Simulation for edges ranging from of 100 to 300 µm have been performed.\n\nFigure 3.19 shows the electron concentration for an unirradiated sensor of different dead\n\nedge widths.\n\nWe can observe in these simulation that there is no correlation between guard ring\n\nbehavior and the width of the zone between the outer guard ring and the cutting edge of\n\nthe sensor. The potential and electric field distribution along the guard ring was found to\n\n93\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\nbe also the same for each model. After irradiation and SCSI, an undepleted zone remains\n\nvisible even for a small edge width of 100 microns. Reduction of the dead edge outside the\n\nguard ring structure appear to be a valid method to reduce the inactive area of the sensor.\n\nFigure 3.20 show the the majority carrier concentration profile at half depth in the three\n\nmodels under study, unirradiated, as a function of distance from the cutting edge. The\n\nlateral depletion limit can be seen for each model as the distance where the concentration\n\nstart to fall rapidly. This figure shows that the lateral depletion seems to be little influenced\n\nby the type of guard ring structure used in a sensor.\n\nFigure 3.20 – Comparison of lateral depletion for the three model of guard ring structure\nunder study\n\nThe guard rings structure itself also represent an inactive area of the sensor and must\n\nbe reduced, The actual ATLAS pixel sensor has been simulated with 0,1,3,4 of the outer\n\nguard ring removed. Figure 3.21 shows the bias voltage distribution taken by the guard\n\nrings for the different simulated structures with an applied bias of 500 V. It is shown that\n\nthe bias voltage of the guard rings are almost the same as before their removal, with the\n\nouter guard rings moving closer to ground while never reaching it in the case where 6 or\n\n10 guard rings were removed. The rest of the transition occur on the surface of the safety\n\n94\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\nedge. Once the guard ring structure get smaller than the lateral depletion zone, it loses its\n\neffectiveness at ensuring a complete transition from high voltage to ground. This limits the\n\nnumber of guard rings that can be safely removed from the actual structure.\n\nFigure 3.21 – Simulated bias voltage distribution for actual ATLAS sensor with 2,4,6 and\n10 of the outer guard rings removed\n\n95\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\nFigure 3.22 – Simulated electric field distribution, 0.1 µm under surface, for actual ATLAS\nsensor with 2,4,6 and 10 of the outer guard rings removed\n\nFigure 3.22 shows the electric field distribution at 0.1 µm under the guard rings for the\n\ndifferent simulated structures with an applied bias of 500 V. The high peak next to outer\n\nguard ring is due to the rapid bias voltage drop from outer guard ring to passivated edge as\n\nthe lateral depletion limit pass the outermost guard ring to extend in the outer edge of the\n\nstructure. Removal of 4 guard rings could be use to reduce the width of the actual structure\n\nas the electric field at the outer of remains small compared to the breakdown electric field in\n\nsilicon, valued at 300 kV/cm. If more guard rings are removed , the breakdown probability\n\nis increased by the incapacity of the guard ring to provide a complete transition from edge\n\nto pixel voltage. The peak value at the outer edge of the guard ring can become big enough\n\nto generate a breakdown and electric field present close to the edge can cause excessive\n\nleakage current.\n\nThinning of a pixel sensor can be beneficial to a detector in high fluence environment.\n\nCharge collection occurs in a small region as trapping time becomes smaller than the drift\n\ntime of the charge deposited in the sensor. Charge deposited deeper in the sensor can never\n\nreach the collecting electrode and do no generate signal, leaving most of the sensor useless.\n\n96\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\nLower bias voltage is needed and trapping is reduced due to small travel distance of holes\n\nand electrons in the bulk in thinner sensors. I performed simulation of the actual ATLAS\n\npixel design with a thickness of 150, 200 and 250 µm. Figure 3.23 shows the electric field\n\ndistribution 1 µm under the pixels. for a bias voltage of 150 V. We can observe that the peak\n\nelectric field at the guard ring, for the same bias voltage, is not affected by the thickness of\n\nthe sensor. This is because the field distribution is mainly related to the lateral depletion\n\ndepth of the sensor. The lateral dimensions of the sensor are the same so the guard ring\n\nbehave similarly in thinner and thicker sensors. The mean electric field inside the sensor is\n\nhowever higher due to reduced thickness. Lateral depletion at half-height, defined as the\n\ndistance from edge where the silicon is undepleted, is 700 µm for the 150 µm thick model,\n\n540 µm for the 200 µm thick model and 480 µm for the 250 µm thick model.\n\nFigure 3.23 – Electric field 1 µm under the guard rings for different ATLAS pixel sensor\nthickness\n\nFor a given constant bias voltage, we notice that the depletion is more complete in\n\nthe thinner sensor as the electric field is higher. This means that a thin sensor could be\n\noperated at lower voltage, with a guard ring structure behaving like in a thicker sensor.\n\nThe breakdown occurring in guard rings are due to high electric field at the junction edge\n\nof the guard rings. Since this electric field distribution is not dependent on thickness, guard\n\n97\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\nring structure are more effective in thinner sensor.\n\n98\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\n(a) ATLAS actual design, n-in-n\n\n(b) Large guard rings, n-in-p design\n\n(c) Small guard rings n-in-p design\n\nFigure 3.15 – Simulated backplane current vs bias potential, for different irradiation doses\n\n99\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\n(a) ATLAS actual design, n-in-n\n\n(b) Large guard rings, n-in-p design\n\n(c) Small guard rings n-in-p design\n\nFigure 3.16 – Simulated guard rings potential vs bias potential, for different irradiation\ndoses. Bias on, high voltage electrode, pixels at 0V\n\n100\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\nFigure 3.17 – Comparison of the electric field at 1 µm under the guard rings for the large\nring and small ring n-in-p model and the ATLAS actual n-in-n model for a bias voltage of\n400V.\n\n101\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\n(a) 100 µm edge width\n\n(b) 200 µm edge width\n\n(c) 300 µm edge width\n\nFigure 3.19 – Simulated electron concentration profile for an unirradiated ATLAS n-in-n\npixel sensor for various dead edge width\n\n102\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\n3.3.3 The Slim Edge Guard Ring structure\n\nIt was proposed by the ATLAS Planar Pixel Sensor upgrade group to reduce further\n\nthe inactive area of the n-in-n sensor by shifting the pixel under the pixel area, as shown\n\nin figure 3.24. This would reduce the area uncovered by pixel and consequently allow much\n\nsmaller inactive edges.\n\nThis structure was simulated to obtain to understand the effects of this shift on the\n\nperformance of the sensor. The model used was the ATLAS n-in-n standard guard ring\n\nshifted under the pixels by 100,200 and 400 µm. The edge was 500 microns but reduction\n\nof the dead edge does not affect the guard ring behavior and the result can be valid as long\n\nas the width of this edge stay within the 100 microns mentioned earlier. Bias Voltage was\n\n150 V and detector thickness was 300 µm. Figure 3.25 show the electric field magnitude\n\nin the region of the edge pixel where overlap occur, for a shift of 100 and 200 µm. The\n\npresence of the guard ring affect the distribution of the field under the pixel and the field\n\nis weaker in the section of the pixel overlapping the guard ring. Figure 3.26a show the\n\npotential distribution in the 200 µm shift case. We can observe that the potential gradient\n\nin the overlap region will drag charge away from the pixel as the drift closer to it. This\n\nis because this part of the pixel is in the lateral depletion zone that is normally located\n\noutside the volume of the edge pixel. This region is the region where the Ramo potential of\n\nthe edge pixel, shown in figure 3.26b, undergo the largest variation. This will lead to loss\n\nin charge collection. This can still be a beneficial method to reduce the edge of the sensor\n\nwhile keeping larger guard ring structures if the signal collected is still sufficient to obtain\n\na good trigger efficiency.\n\nThe big advantage of the slim edge structure comes after irradiation. After SCSI, the\n\ndepletion occurs from the pixel side and the guard ring become less efficient. The electric\n\nfield distribution at the edge pixel is then changed and the problem of the lateral depletion\n\nzone go away. Figure 3.27 show the electric field configuration in the 200 µm shift model\n\nafter a dose of 1015neqcm\n−2 at a bias voltage of 1000 V.\n\nAs the detector at IBL and SLHC will be during its lifetime irradiated, this strategy is\n\na good compromise to obtain a slim edge structure meeting the requirements of the IBL\n\n103\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\nFigure 3.24 – Slim edge Multi Guard Ring structure showing the overlap of the pixel with\nthe guard rings on the backside\n\nwhile keeping a large Multi-Guard Ring Structure. This guard ring structure was proposed\n\nfor the sensor candidate sensor for the IBL project. The shift is 250 µm in the case of the\n\nso called \"Slim-Edge\" candidate , and 100 µm in the case of the conservative candidate.\n\nTest structure with FE-I3 geometry were produced in the PPSU09 sensor production and\n\nwere used in beam test to study the properties of the edge pixels. Results will be presented\n\nin the next chapter. The simulation made an important prediction on the behavior of\n\nthe slim edge structure that were demonstrated in experiment : The loss of charge at the\n\nedge in the overlap zone between the pixel and guard rings. This phenomenon was later\n\nobserved experimentally. The work done on TCAD simulation of guard ring structure has\n\ncontributed to the design of the various planar pixel structure proposed for the upgrade\n\nprojects. The simulation provided guidelines on the design change that would affect or not\n\nthe performance of the detector. In the next chapter, a comparison of TCAD results with\n\nexperimental data will show the good performance of the simulation in predicting device\n\nbehavior before and after irradiation. Monte-Carlo Simulation of the charge transport was\n\nalso performed to replicate the charge collection behavior observed experimentally.\n\nFor the detector irradiated at super LHC fluence (5− 10× 1015neqcm\n−2), new physical\n\nphenomenon we observed experimentally. In the next section, I present a model that was\n\nelaborated using TCAD simulation to emulate the behavior of highly irradiated sensors.\n\n104\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\n(a) 100 µm shift\n\n(b) 200 µm shift\n\nFigure 3.25 – Electric field magnitude in Slim edge Multi-Guard ring structure with a\nshift of 100 and 200 µm\n\n105\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\n(a) Potential distribution\n\n(b) Ramo Potential distribution\n\nFigure 3.26 – Potential and Ramo Potential distribution in a Slim edge Multi-Guard Ring\nstructure with a shift of 200 µm. Arrows show the drift direction of electrons\n\n106\n\n\n\n3.3. THE MULTI-GUARD RING STRUCTURE\n\n(a) Potential distribution\n\n(b) Electrifc field distribution\n\nFigure 3.27 – Potential and Electric field distribution in a Slim edge Multi-Guard Ring\nstructure with a shift of 200 µm after an exposition to a fluence of 1015neqcm\n\n−2 . Arrows\nshow the drift direction of electrons\n\n107\n\n\n\n3.4. THE CHARGE AMPLIFICATION MECHANISM IN HIGHLY IRRADIATED\nSILICON SENSORS\n\n3.4 The charge amplification mechanism in highly irradiated\nsilicon sensors\n\nCharge collection efficiency (CCE) of silicon planar pixel sensors used for charged par-\n\nticle detection in high energy physics is known to be reduced by increased exposition to\n\nradiation damage. However it has been observed experimentally [47; 48] that CCE of pla-\n\nnar pixel sensors exposed to fluences of the order of 1015−16 neqcm\n−2 can be increased\n\nby applying higher bias voltage to the sensor. Figure 3.28 show an example of unexpec-\n\nted charge collection in 80 µm pitch, 140 and 300 µm thick n-in-p strip sensors. In this\n\nsection , I present a set of TCAD simulations that have been performed to explore the\n\npossible mechanisms behind this anomalous charge collection observed after exposition to\n\nhigh fluences of diode and strip sensors.\n\nFigure 3.28 – Experimental charge collection in ir-\nradiated thin and thick n-in-p strip sensors showing\nevidence of charge amplification [47]\n\nThe simulations were perfor-\n\nmed on simple geometries to ex-\n\nplore the effects of impact ioniza-\n\ntion and trap-to-band tunneling on\n\nthe transient behaviors of planar\n\nsilicon sensors. A simple one di-\n\nmensional diode geometry was si-\n\nmulated. Implants parameters used\n\nwere obtained from measurements\n\nfrom the PPSU09 production. Two\n\ndimensional simulation of a strip\n\nsensor was also performed but the\n\ncomputing time required to per-\n\nform a transient simulation was too\n\nlarge to obtain a large number of simulation in a reasonable time. For numerical accuracy,\n\nthe 1D simulation considers a width of 250 µm with laterally uniform conditions. The\n\nTCAD simulation have shown poor convergence when aspect ration of the structure is far\n\nfrom unity. DC simulation for a fluence of 2×1016 neqcm\n−2 for a bias voltage up to 3000 V,\n\nwhen numerically possible, was performed for a thick (300 µm) and thin (140 µm) sensor,\n\n108\n\n\n\n3.4. THE CHARGE AMPLIFICATION MECHANISM IN HIGHLY IRRADIATED\nSILICON SENSORS\n\nwith impact ionization switched on and off, then with trap-to-band tunneling switched on\n\nand off. This allows to see the effect of each mechanism on the behavior of the sensors.\n\nTransient simulation of the device was also performed. A 2 ns triangular 1060 nm laser\n\npulse was sent perpendicular to the device surface. The TCAD software then calculates\n\nthe ionization by the laser, mostly uniformly deposited in the bulk depth, and perform\n\na transient simulation over 40 ns at different bias with the trap-to-band tunneling and\n\nthe impact ionization turned on and off. The signal we obtain from the simulation is then\n\nintegrated, with the pedestal subtracted, to obtain the collected charge. From simulation,\n\nwe also obtained the initially deposited charge and we can then compute the CCE using\n\nequation (3.7).\n\nCCE =\nQcollected\nQdeposited\n\n(3.7)\n\nThe DC simulations were performed on a non-irradiated diode, then on a heavily ir-\n\nradiated diode to a fluence of 2 × 1016 neqcm\n−2. Fig. 3.29 shows the current circulating\n\nin the high voltage electrode for a thick and a thin diode with the impact ionization and\n\ntrap-to-band tunneling successively turned off in the simulations. This allows us to see the\n\ncontribution of the different physical phenomena to the DC characteristics of the sensors.\n\nRecombination lifetime was set to 1× 10−5 s for a better simulation of leakage current in\n\nhigh resistivity silicon used for pixel sensors [49].\n\nThis parameter is not taken into account in irradiated diode simulation as it is overrid-\n\nden by the defect introduction model . Fig. 3.29(a) shows the IV curve before irradiation.\n\nA hard breakdown occurs at high voltage with a steep increase in leakage current. The\n\ncontribution to leakage of impact ionization is negligible before breakdown, explaining why\n\nno charge multiplication is observed in non-irradiated sensors. The breakdown occurring in\n\nthe diode is located inside the bulk under the implant, as electric field there becomes too\n\nelevated, as can be seen in the pre-breakdown electric field profile in Fig. 3.30(a), causing\n\nhole multiplication to occurs. Since this is a one dimensional simulation, no implant edge\n\neffects are considered. The contribution of trap-to-band tunneling to leakage current in the\n\ndiode becomes important even for low bias voltage. As electric field increases in the bulk\n\n109\n\n\n\n3.4. THE CHARGE AMPLIFICATION MECHANISM IN HIGHLY IRRADIATED\nSILICON SENSORS\n\nof the sensor , this mechanism contribute to more and more of the leakage leading to this\n\nresistive behavior after depletion. Fig. 3.29(b) shows the IV curve for the same sensors\n\nafter irradiation. The contribution of trap-to-band tunneling to leakage current becomes\n\nsignificant at higher bias voltages than before irradiation. The breakdown occurring in the\n\nsensors before irradiation is replaced by a soft breakdown where the current rises at a much\n\nlower rate even if the same high field region is observed in the thin diode electric field profile\n\nin fig. 3.30(b). This allows the operation of the sensor in this regime where multiplication\n\neffects are to be expected. This quenching of the avalanche mechanism can be explained by\n\nthe increased trapping in irradiated sensors, where the mean free path of holes is reduced.\n\nThis prevents the free holes of creating an avalanche and reduces the contribution from\n\nimpact ionization to leakage current.\n\n110\n\n\n\n3.4. THE CHARGE AMPLIFICATION MECHANISM IN HIGHLY IRRADIATED\nSILICON SENSORS\n\n(a) Current versus bias voltage, non-irradiated 1D diode\n\n(b) Current versus bias voltage, 1D diode φeq = 2× 1016 neqcm\n−2\n\nFigure 3.29 – Comparison of Current versus Voltage curves before and after irradiation.\n\nThe results of the transient simulation can be seen in fig. 3.31. The multiplication\n\neffect can be seen only at high voltage in non-irradiated sensors (fig.3.31(a)), much beyond\n\nthe breakdown voltage. Charge collection follows the normal behavior for non-irradiated\n\nsensors, saturation at a CCE of 1. The multiplication effect in irradiated sensors can be seen\n\nin fig. 3.31(b). Trapping should normally prevent some charge of being collected. A small\n\n111\n\n\n\n3.4. THE CHARGE AMPLIFICATION MECHANISM IN HIGHLY IRRADIATED\nSILICON SENSORS\n\npart of the deposited charge is recovered by trap-to-band tunneling and impact ionization\n\nfurther improves the charge collection efficiency to a factor superior to 1. The electric field\n\nprofile at a bias in the multiplication regime, as seen in fig. 3.30(b) explains the presence of\n\nsuch effect in the irradiated sensor. In this figure is represented the electric field in the bulk\n\nof the diodes at a bias voltage close to breakdown, for non-irradiated sensors, and in the\n\nmultiplication regime for irradiated sensors. A high electric field, of the order of 100 kV/cm,\n\nexists in the bulk of the irradiated sensors leading to de-trapping and multiplication in the\n\nbulk , as seen in fig. 3.31(b). It should be noted that de-trapping alone cannot explain\n\ncharge collection efficency over 1. For the non-irradiated diode, the electric field profile\n\nshown in fig. 3.30(a) shows the pinching if the electric field close to the readout implant\n\nthat eventually cause a hard breakdown , as seen in fig. 3.29(a), at higher bias voltage.\n\nThis breakdown prohibits the operation of non-irradiated diodes at such bias voltage. In\n\ncase of irradiated diodes, the attenuation of impact ionization by trapping allows operation\n\nat higher voltage allowing to reach the multiplication regime.\n\n112\n\n\n\n3.4. THE CHARGE AMPLIFICATION MECHANISM IN HIGHLY IRRADIATED\nSILICON SENSORS\n\n(a) Electric field at bias before breakdown, non-irradiated diodes\n\n(b) Electric field at a bias voltage in the multiplication regime, irradiated\ndiodes φeq = 2× 1016 neqcm\n\n−2\n\nFigure 3.30 – Comparison of Electric field magnitude at a possible operation bias voltage\nbefore and after irradiation.\n\n113\n\n\n\n3.4. THE CHARGE AMPLIFICATION MECHANISM IN HIGHLY IRRADIATED\nSILICON SENSORS\n\n(a) CCE versus bias voltage, unirradiated 1D diode\n\n(b) CCE versus bias voltage, 1D diode φeq = 2× 1016 neqcm\n−2\n\nFigure 3.31 – Comparison of CCE curves before and after irradiation.\n\nThe simulation performed demonstrates that the parametrization of radiation damage\n\nin terms of effective defects introduction, combined with trap-to-band tunneling and im-\n\npact ionization qualitatively explains the charge recovery effect observed experimentally\n\nin highly irradiated n-in-p diodes. The high electric field present at the readout implant\n\ncausing multiplication of free charges, combined with increased de-trapping caused by the\n\n114\n\n\n\n3.4. THE CHARGE AMPLIFICATION MECHANISM IN HIGHLY IRRADIATED\nSILICON SENSORS\n\nbending of the band gap structure by the electric field and the attenuation of the multipli-\n\ncation by increased trapping provides a physical explanation to the observed behavior of\n\nthese diodes.\n\n115\n\n\n\n3.4. THE CHARGE AMPLIFICATION MECHANISM IN HIGHLY IRRADIATED\nSILICON SENSORS\n\n116\n\n\n\nChapitre 4\n\nFrom TCAD simulation to\nexperimental data\n\nTCAD simulation models presented in the last chapter require the input of a large\n\nnumber of parameters to obtain quantitatively comparable results. Process parameters de-\n\ntermine the shape and concentration profile of the implants forming the guard rings and\n\nthe readout electrodes. Quality of the silicon dioxide and silicon interface is represented\n\nby the surface charge used. The resistivity of the bulk and recombination lifetime of the\n\ndefects present in the bulk affect depletion potential and leakage current magnitude. Ex-\n\nperimentation on test structure and sensors can help to obtain the parameters needed to\n\ntune the simulation models and obtain quantitative results.\n\nAccurate TCAD simulation can offer better understanding of the behavior of the AT-\n\nLAS existing and future pixel sensors. The inclusion of radiation damage behavior and\n\nedge effects using knowledge from TCAD simulation in the digitization of pixel sensors\n\nused in the ATHENA simulation of the ATLAS detector can improve the realism of the\n\nsimulation and help understand effects observed during the lifetime of the inner detector.\n\nA Monte-Carlo charge transport code was built to study transient behavior of sensor using\n\nTCAD electric field and Ramo potential simulation as input and provide more information\n\non sensor detection properties. As mentioned in the previous chapter, a sensor production\n\nincluding designed structure influenced by our TCAD simulation presented in this work\n\nwas delivered to us in 2010. This production included many test structures to study our\n\nTCAD models. In this chapter, I present the experimental work that was performed to\n\n117\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\ncalibrate the simulation models used in this work. Part of the work behind this has been to\n\nassemble test bench for low noise,low current and high voltage DC and transient characte-\n\nrization of silicon sensors. Clean room and test beam measurements were used to calibrate\n\nthe parameters used in simulation and get more knowledge on the physics describing guard\n\nrings structure.\n\nThe PPSU09 production contained ATLAS pixel structure with FE-I3 geometry with\n\nmany variations of slim edges structure and reduced number of guard rings. These sen-\n\nsors were bonded to FE-I3 chip and tested before and after irradiation with high energy\n\nprotons and neutrons in test beam at CERN SPS and DESY electron synchrotron. The\n\nEUDET telescope was used to provide track information used to analyze the pixel sensor\n\ndata. Results from reconstruction and analysis of these sensor have been used to validate\n\npredictions done using TCAD simulation. Finally , a digitization model for the FE-I3 and\n\nFE-I4 planar pixel sensors candidate for IBL was developed using knowledge gathered from\n\nGEANT4 Monte-Carlo and TCAD simulation. The GEANT4 simulation of the EUDET\n\ntelescope setup was used to simulate test beam conditions including digitization method\n\nand was used to validate the model. Comparison of test beam data and simulation data\n\nshow good agreements and reproduce well edge and radiation damage effects that will\n\noccurin the ATLAS detector after long operation time.\n\n4.1 Experimental validation of TCAD simulation\n\nThe PPSU09 production contained a set of test structures that we included for to cali-\n\nbration of TCAD simulation models. Table 4.1 show the structure and their experimental\n\npurpose and figure ?? show the layout of the wafer for the n-in-n production.\n\n118\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nFigure 4.1 – N-in-N PPSU09 production wafer\n\n(a) doping profile structure (b) Inter-pixel capacitance\nstructure\n\n(c) LAL diodes\n\nFigure 4.2 – Test structure inserted in the PPSU09 production\n\nThe main goal of the production was to produce prototype of sensors with reduced edge\n\nfor the IBL ATLAS pixel sensor. Several FEI3 pixel structure with guard ring removed were\n\ninserted in the production. The main detector occupying most of the wafer is a FEI4 sensor.\n\nThe n-in-p production contained :\n\n– 3 FE-I3 standard small guard ring design , p-spray\n\n– 3 FE-I3 standard small guard ring design, moderated p-spray\n\n– 3 FE-I3 small guard ring design, 8 guard rings, p-spray\n\n– 3 FE-I3 small guard ring design, 15 guard rings, p-spray\n\n119\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nTable 4.1 – Test structure inserted in PPSU09 production\n\nStructure Description Purpose\nDoping profile\nstructure\n\nThree 1.9 mm × 8.9 mm\nzones with n+, p-spray\nand moderated p-spray\nimplants on front side and\np+ implant on backside.\nFigure 4.2a\n\nAtomic Force Microscopy,\nSpreading Resistance Pro-\nfiling (SRP) and Secon-\ndary Ion Mass Spectro-\nscopy (SIMS).\n\nInter-Pixel Capa-\ncitance structure\n\n5×5 pixel matrix surroun-\nded by a standard guard\nring structure. First neigh-\nbors of the central pixel\nare connected together and\nindependently the second\nneighbor also. Pixel size of\n50, 250 and 400 µm have\nbeen used. Figure 4.2b\n\nInter-pixel capacitance\nmeasurement in clean\nroom.\n\nLAL diodes 6× 6 mm diodes with 1 to\n4 large contactable guard\nrings. Figure 4.2c\n\nLeakage current and guard\nring potential measure-\nment before and after irra-\ndiation.\n\n– 1 FE-I4 small guard ring design, p-spray\n\n– 1FE-I4 small guard ring design, moderated p-spray\n\nThe n-in-n production contained :\n\n– FE-I3 ATLAS standard guard ring with 3,5,11 and 13 outer guard ring removed\n\n– FE-I3 ATLAS standard guard ring with pixel shifted stepwise by 50 microns in 8\n\nsteps\n\n– FE-I3 ATLAS standard guard ring with pixel shifted by 100 and 200 µm\n\n– FE-I4 4 chip module with ATLAS standard guard ring\n\n– FE-I4 ATLAS standard guard ring\n\nA second n-in-n production was submitted to produce IBL planar sensors. Two FE-\n\nI4 sensor were used in this production. The conservative model has the ATLAS standard\n\nguard ring structure with the 3 outer guard ring removed, as suggested in last chapter. The\n\nSlim Edges design has also reduced guard ring, but pixel are also shifted 250 microns under\n\nthe guard rings. The measurement presented in the next section were performed using the\n\n120\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nstructure from these productions.\n\n4.1.1 Doping profile measurements\n\nFigure 4.3 – Spreading Resistance Profiling\ntechnique\n\nTCAD simulation results such as guard\n\nring potential and breakdown voltage are\n\ndependent on the doping profile of the struc-\n\nture. Knowing the doping profile of the dif-\n\nferent implant is therefore an important step\n\nto validate the simulation model and obtain\n\naccurate simulation. Various method exist\n\nto obtain the dopant concentration profile of\n\nan implant. The method can be divided in\n\nthe stoichiometric and the electric methods.\n\nThe stoichiometric methods consist in mea-\n\nsuring the total amount of dopant in the si-\n\nlicon bulk. One method used is Secondary\n\nIon Mass Spectroscopy. It consists in sputtering the surface of a sample using an ion beam\n\nand analyze the ejected secondary ions using a mass spectrometer. The speed at which the\n\nion beam dig in the sample is known and the amount of ions of each species measured in\n\nthe spectrometer as a function of time can be converted into a concentration versus depth\n\nprofile. The SIMS doping profile were performed at CNRS-Meudon.\n\nTwo electrical methods were also used to characterize the implants. Electrical methods\n\nmeasure the carrier concentration in the implant, which is related to the electrically active\n\ndopant concentration. The first method used was Spreading Resistance profiling. This\n\nmethod consist in probing the local resistance between two very close point on the implant\n\nusing small tungsten needles. The implanted structure is beveled with a small angle to\n\nreveal the implant at different depth. The needle probing is done at different points along\n\nthe bevel to measure the local resistance between the two needle. The resistivity (ρ) of\n\nsilicon vary with active dopant concentration hence the measure of the Resistance R can\n\nbe converted into a active dopant profile. The distance from the surface on the beveled\n\n121\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nplane can be converted to depth of the implant by multiplying the distance by the sinus of\n\nthe bevel angle. Figure 4.3 show the principle of the measurement. The SRP measurements\n\nwere performed by Evans Analytical Group. The second method that was investigated and\n\ndeveloped was the Atomic Force Microscope Spreading Resistance Profiling (AFMSRP).\n\nThis method is analogous to SRP done with the cantilever of an atomic for microscope\n\nthat is electrically connected to an electrometer.\n\nTable 4.2 – Comparison of SIMS and SRP method\n\nMethod SRP SIMS\nResolution 2% 5%\n\nSensibility (cm−3) 1012 − 1020 1015 − 1020\n\nSample size several mm2 >1 mm2\n\nThe measurement were performed on the doping profile structure of the PPSU09 n-\n\nin-n and n-in-p production. This structure was designed to allow many kind of doping\n\nprofile measurement methods on the same test structure. Three zone representing the\n\nreadout n+ implant and the two p+ implant, one representing the p-spray implant and\n\nthe other representing the non-moderated part of the moderated p-spray. The backside of\n\nthe structure is covered by a p+ implant.\n\n4.1.1.1 n-in-n production\n\nTwo samples , labeled 02-42 and 05-42, were tested using both SRP and SIMS method.\n\nfigure 4.4 show to profile that were obtained from the SRP and SIMS method. The n+\n\nimplant corresponding to the pixels of the n-in-n sensors is shown in figure 4.4a. Measure-\n\nment on both sample show the same profile showing a good uniformity of the phosphorus\n\nimplantation on the different wafers of the production. Bulk effective dopant concentration\n\nof the sample can be determined from the concentration of active dopant far from the junc-\n\ntion in the SRP measurement. For the n-in-n wafer sample tested, Neff = 4.15×1012cm−3\n\nfor sample 05-42 and Neff = 1.15×1012cm−3. Using equation 2.5 and 2.6, we can compute\n\nthe expected depletion potential of a 285 µm sensor and compute its resistivity. Sample\n\n02-42 has a resistivity of 4 kΩcm−1 and a depletion voltage of 80V while sample 05-45 has\n\n122\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\na resistivity of 2.2 kΩcm−1 and a depletion voltage of 291V. The profile might however not\n\nhave been performed deep enough to obtain bulk resistivity.\n\nThe p+ implant measurements, corresponding to the backside and guard ring implants\n\nin the n-in-n sensors, are shown in figure 4.4b. The pn junction is clearly visible by the\n\ndrop in active dopant effective concentration as the silicon transition from p to n type. The\n\ndepleted region of the junction is characterized by a low concentration as p and n type\n\ndopant compensate each other leading to a quasi null effective dopant concentration and\n\nvery high resistivity. junction can be found at 582 nm for sample 02-42 and at 676 nm for\n\nsample 05-42. An order of magnitude is seen between the activated and total concentration\n\nof boron present in the implant possibly pointing to and incomplete activation of the\n\nimplant or a compensating defect present in the bulk lowering the effective active dopant\n\nconcentration. The effective active dopant for sample 02-42 in this measurement is Neff =\n\n1.15 × 1012cm−3 for sample 02-42 and Neff = 1.85 × 1013cm−3 for sample 05-42. This\n\nyield to a resistivity of 248.4 Ωcm−1 and an unrealistic depletion voltage of 1300 V. This\n\nmean that doping profile was not performed deep enough to obtain bulk resistivity which\n\nis expected to be over 1000 Ωcm−1. For sample 05-42, we obtain a bulk concentration of\n\nNeff = 1.15× 1012cm−3 leading to a a resistivity of 4 kΩcm−1 and a depletion voltage of\n\n80V, compatible with sample 02-42 for the n implant measurement.\n\n123\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\n(a) N+ implant\n\n(b) P+ implant\n\nFigure 4.4 – N+ and P+ Doping profiles measurement on the n-in-n samples of the\nPPSU09 production\n\nP-spray and moderated p-spray implantation profiles were also measured on these test\n\nstructure. The moderated p-spray correspond to a shallow low concentration boron implant\n\nwhile the p-spray correspond to a deeper implant with slightly higher concentration. The\n\ndifference between both profiles, originationg from the same implantation of boron on the\n\nfrontside of the wafer is the nitride layer that is used as a filter that is present for the\n\ncase of low p-spray implant. Figure 4.5a and 4.5b show the two implant profiles measured\n\nby the SRP and SIMS method. SIMS measurement are uniform between sample and the\n\n124\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\npeak concentration of the p-spray implant is shifted with regard to the moderated p-\n\nspray implant as expected because of the presence of the additional nitride layer during\n\nimplantation. SRP measurement, however, show no sign of a junction for all measurements\n\nexcept for the p-spray of sample 02-42. As boron presence is confirmed by SIMS, it is highly\n\nprobable that the activation step of the p-spray implants was not completed correctly\n\nduring the processing of the diodes . No p-spray was used in the simulation of the n-in-n\n\nsensor although it was implemented as a possible step in the simulation template.\n\n4.1.1.2 n-in-p samples\n\nThe same measurements shown in last section were performed on a doping profile test\n\nstructure and a standard diode of the n-in-p PPSU09 production from the same wafer.\n\nFigure 4.6a show the n implant representing the pixel and guard rings implants of the\n\nn-in-p pixel sensors. The SIMS profile confirm the presence of a phosphorous layer in the\n\nimplant and the measurement are compatible with the measurement performed on the\n\ndiode’s n implant. SRP measurement show that activating of the implant was completely\n\ncorrectly. However, no junction is found in the profile. A junction is however found on\n\nthe backside of the sample as show in figure 4.6b. As the bulk is expected to be p-type,\n\nposition of the junction is incorrectly located in the device. This lead us to suspect the\n\npresence of an additional donor in the bulk of the sensor that inverts the type of the bulk\n\nin the manner of radiation damage. It was found that the wafers from were originated\n\nthese sample was subjected to an oxygenation at high temperature to increase its radiation\n\nhardness. The heating of the oxygenated silicon at temperature between 300 and 550 C\n\nwill lead to the formation of thermal donors [50]. While thermal donor are annealed at\n\nhigher temperature used for oxidation, if cooling of the wafer back to to room temperature\n\nis not done sufficiently fast, formation of thermal donors could have occurred during the\n\ntransition. These thermal donor can eventually be sufficiently abundant to invert the type\n\nof the silicon bulk. Further SRP measurements were performed on the diode structure\n\nto determine how deep in the bulk the substrate was inverted. Figure 4.8 show the SRP\n\nmeasurements made on the sample diode on the first 15 microns of the implant. The\n\nmeasurements were performed down to 140 µm under the p and n implants. A junction\n\n125\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\n(a) moderated p-spray implant\n\n(b) p-spray implant\n\nFigure 4.5 – Insulation doping profiles measurement on the n-in-n samples of the PPSU09\nproduction\n\n126\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nwas found on the backside but in not shown in the figure as stepping was too large. The\n\neffective dopant concentration found at the center of the diode was Neff = 2.6×1011cm−3,\n\ncorresponding to a n-type resistivity of 17.6 kΩcm−1 and a depletion voltage of 18 V\n\nconsidering a 300 µm thick sensor.\n\n(a) N+ implant\n\n(b) P+ implant\n\nFigure 4.6 – N+ and P+ Doping profiles measurement on the n-in-p samples of the\nPPSU09 production\n\nThe p-spray and moderated p-spray implantation are extremely important for the n-in-\n\np sensors as the guard ring structure is located on the electron collection surface. Figure 4.7\n\nshow the doping profile measure on the samples for these implants. The SIMS measurement\n\n127\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nconfirm the presence of boron and the SRP measurement show a well activated profile. The\n\nmoderated p-spray profile show a shift towards the left with regard to the p-spray profile,\n\nas expected. A junction can be found in both SRP profile confirming again the inversion\n\nof the bulk type due to thermal donor generation.\n\n(a) moderated p-spray implant\n\n(b) p-spray implant\n\nFigure 4.7 – Insulation doping profiles measurement on the n-in-p samples of the PPSU09\nproduction\n\nNew sensor production taking into account the results from our measurement were\n\nplanned to fix the problems encounter in the process. Independent measurements using\n\n128\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nFigure 4.8 – SRP measurement performed deep into a n-in-p diode sample\n\nSIMS was performed by LPHNE laboratory and confirm the results presented here. While\n\nSIMS and SRP represent efficient methods to characterize the implants of a production, the\n\ncost of each measurements limits the quantity that can be performed for each production.\n\nin the next section, I present a method that is in development at LAL to perform electrical\n\nprofile measurement using our own setup.\n\n4.1.1.3 Atomic Force Microscopy Spreading Resistance Profiling\n\nAtomic Force Microscopy (AFM) measure the repulsive and attractive forces between\n\na nanometer scale cantilever and the surface atoms of a sample. Figure 4.9 show the sche-\n\nmatics of a typical AFM measurement. The cantilever tip’s distance from the surface is\n\nmeasured using the deflection of a laser ray reflecting on its back surface. A very accurate\n\npiezoelectric translation system is used to scan sample surface with the cantilever and ob-\n\ntain a measurement of the topology of a surface. This method allow to study nanoscale\n\nfeature of a surface and allow to perform accurate measurements while scanning on short\n\ndistances. The Institut d’Electronique Fondamentale (IEF) located on Orsay campus is\n\nequipped with an AFM equipped with a resiscope. The resiscope is used to measure the\n\nelectrical resistance between the tip of the needle and the backside of a sample when a\n\ncertain bias voltage is applied. This apparatus allows us to scan and measure local resisti-\n\n129\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nvity of beveled implant structures by scanning the surface of the bevel and measuring local\n\nresistivity. Calibration sample with known resistivity can be used to calibrate the method\n\nand translate resistance measurements into carrier concentration profiles.\n\nFigure 4.9 – Atomic force micro-\nscope schematics\n\nIn order to proceed with the measurement, a be-\n\nvel of a small known angle had to be etched from the\n\ndoping profile test structure. IEF own a polishing ma-\n\nchine,a MECAPOL P400, that can be used to etch\n\nand polish the surface of our sample. A mechanical\n\nstructure was design by Tristan Vandenberghe, me-\n\nchanical engineer at LAL, to allow to polish the small\n\nsilicon test structure. Figure 4.10 show the CAD dra-\n\nwing of the polishing device.\n\nFigure 4.10 – Sample holder designed for small angle beveling in doping test structure\nusing the MECAPOL P400 polishing machine.\n\nFigure 4.10 show a bottom view of the device\n\nshowing the sample holder with a doping profile test\n\nstructure fixed on the angle surface using double si-\n\nded conductive tape. The beveling of the surface is performed by first removing the extra\n\n130\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nlayer of silicon from the sample using a 2400 grain per square inch abrasive paper. The\n\nsurface of the bevel is then polished using successively small grain polishing liquid with\n\ngrain size of 9,3 and 1 µm. A final polishing is done using a commercial colloidal silica\n\nsolution named NALCO. Through the polishing process, the rugosity of the surface has\n\nbeen monitored using a profilometer throughout each step to adjust the step of the poli-\n\nshing protocol to develop a standardized procedure. Optical inspection of the sample has\n\nalso been performed after each polishment. Figure 4.11 show the evolution of the typical\n\nprofile of the surface after various step of the process. It can be observed that the variation\n\nin height of the surface is reduced after each polishing step. The final state of the surface\n\nshould exhibit defects of a typical size of 1 µm or less. The final angle of the beveled\n\nstructure was measured by mechanical profilometry to allow for correct translation from\n\nmeasured distance from bevel edge to doping profile depth.\n\nFigure 4.11 – Evolution of rugosity of a doping profile test structure after successive\npolishing steps\n\nAt the moment of writing these lines, several problem have been observed in the po-\n\nlishing procedure. The problems range from wrong polishing time to incompatibility of\n\ncertain substance present at the surface of certain sample, such as the BCB used to insu-\n\nlate the guard rings of the n-in-p sensor, with the polishing liquids. However, preliminary\n\n131\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nresults obtained on two samples shown the feasibility of the method and its ability to\n\nmeasure small implant structure. Raw results of measurements performed on the n-in-p\n\ntest structure are shown in figure 4.12. The results show the resistance of the surface as a\n\nfunction of the position of the scanning cantilever. The resistance is expressed in terms of\n\nthe raw output of the resiscope in V, ranging from 0 to 10 V corresponding to a resistance\n\nrange of 102 to 1012 Ω. Calibration sample will be measured to obtain an absolute value of\n\nthe carrier concentration. Knowing the bevel angle, measured with mechanical profilome-\n\ntry, we can obtain the actual doping profile of the implant by multiplying the x distance\n\nby the sinus of the bevel angle. The relative depth of the implant is in agreement with the\n\npreviously measured value for this structure.\n\n(a) n+ implant (b) moderated p-spray implant (c) p-spray implant\n\nFigure 4.12 – Raw image of the resiscope signal for a scan of a 50x50 µm surface. Right\nside of the figure represent the surface of the structure.\n\nFigure 4.13 show the converted doping profile of the n implant of figure 4.12, averaged in\n\nthe X direction. The calibration between resistance and concentration values was done using\n\npreliminary measurements done on calibration sample with fixed carrier concentration. The\n\nvery good agreement between the preliminary results obtained and the other doping profile\n\nmeasurement gives us confidence in the reliability of the method and further step will be\n\ntaken to standardize the method to obtain reproducible results.\n\nAnother advantage of the AFM method is that it allows us to get two dimensional image\n\nof the implants if the implant lateral dimension can be constrained within the scanning\n\nwindow of the AFM, usually 50 x 50 µm. Trials have been performed to measure the p+\n\nimplant of the ATLAS n-in-n guard ring structure. A FE-I3 sensor was cut to expose the\n\nguard ring structure. The surface of the guard rings was beveled to expose the implants\n\n132\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nFigure 4.13 – Comparison between the doping profile of the n implant of the n-in-p\nproduction with the SRP and SIMS measurements performed on the sample\n\nand an AFM measurement was performed on the sample. Figure 4.14 show the converted\n\nprofile obtained on the sample. The surface of the sample, covered by oxide and nitride, can\n\nbe seen in the lower part of the plot where resiscope output values are very high. This high\n\nresistance is due to the bad contact between the cantilever and the oxide. Two complete\n\nguard ring implants of a 10 µm width can be seen, as expected for this structure. The\n\ndepth of the implants is compatible with the measured depth for the n-in-p production.\n\nFurther work is needed to obtain more accurate and less noisy measurement. However, the\n\npreliminary results obtained and presented here have demonstrated the feasibility of the\n\nmethod. No other doping profile measurement methods allow for two dimensional profiling\n\nof the implants. This method makes it a very interesting candidate for comparison with\n\nTCAD simulation model of implantation required to create the structure used in electrical\n\nsimulation.\n\n4.1.1.4 Calibration of TCAD implantation model with experimental data\n\nWith the data of the PPSU09 doping profile structures, we were able to determine\n\nsome of the main missing parameters needed for accurate simulation of the implantation in\n\n133\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\n(a) SPR Raw measurement\n\n(b) Photography of the measured region\n\nFigure 4.14 – 2D doping profile of an n-in-n ATLAS standard guard ring structure\n\n134\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nTCAD process simulation. As p-spray and moderated p-spray were shown to have problems\n\nin the production, the experimental profile were used directly in the simulation to avoid\n\nthe complex problem of finding the process step leading to such profile. Same method was\n\ndone for simulation of the n-in-p sensor’s n and p implant. The presence of thermal donor\n\naffect the doping profile and makes it very hard to disentangle the different process step\n\nleading to such profile .\n\nApart from the lack of activation of the p-spray for some sample, the n-in-n sample have\n\nshown the expected doping profile characteristics. The parameters needed to reproduce the\n\nimplantation process of these implants are the temperature and duration of the thermal\n\nannealing used to activate the dopants, the thickness of the oxide layer used a screen and\n\nthe energy and dose of the implanted ions. The Modified Levenberg-Marquart algorithm\n\n[39], built-in SILVACO software, was used to obtain the value of these parameters that\n\nproduce a simulation of the doping profile that most closely fits the experimental SIMS\n\nand SRP measurements.\n\nTwo models are available in SILVACO TCAD software to simulate the diffusion of the\n\ndopants : The fermi diffusion model and the fully coupled diffusion model . The fully coupled\n\nmodel includes the physics of the previous model and adds a new phenomenon needed for\n\nmore precise simulation of small details of the doping profile. The physics behind each of\n\nthe model is described in the TCAD software manual [39]. Two implantation models are\n\nalso available. A simple model which consist of a parametrization of implant profiles using\n\nvarious SIMS measurements can be used for fast simulation. A Monte-Carlo model using\n\nthe binary collision approximation, can be used to obtain a more accurate description of\n\nthe implant. However, the computing time needed increases by an order of magnitude when\n\nusing the more accurate model. This limits the usability of the two last models for large\n\nstructure simulation.\n\nOptimization of the implantation parameters was performed using the simple diffusion\n\nand implantation models. The advanced models were then activated to compare the ob-\n\ntained profile to the SIMS measurements. figure 4.15 show the comparison between the\n\ndoping profile obtained and the SIMS profile for the n-in-n p and n implants. It can be\n\nobserved that the simulated profile reproduce well the shape of the experimental data. As\n\n135\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nmore complex model are used, even more accurate description of the profile is obtained.\n\nParameters used to produce these profiles are shown in table 4.3. These parameters we\n\nused for the n-in-n sensor simulation shown in the last chapter. Experimental profiles were\n\ndirectly injected in the simulation in the case of the p-spray implants. Simulation of the\n\nn-in-p structure were also performed using the experimental data when available.\n\nTable 4.3 – Optimized doping and diffusion parameters for the n-in-n PPSU09 production\n\nParameter N+ implant P+ implant\nOxide thickness (nm) 40 40\n\nImplantation dose (cm−3) 1.38e15 5.5e14\nImplantation energy (keV) 10 10\nAnnealing temperature (C) 1350 1293\nAnnealing duration (min) 0.22 0.15\n\n4.1.2 Guard Ring measurements\n\nThe guard ring structure of the planar pixel sensor of the PPSU09 production have been\n\nsimulated to predict their behavior under bias. The main measurable parameter is the bias\n\nvoltage taken by the guard ring when the sensor is biased. In the clean room, a setup\n\nusing a probe station enclosed in a copper faraday cage, connected to a Keithley 6517B\n\nhigh voltage source and an electrometer was used to measure the potential distribution\n\nof various guard ring structures. An electrometer with a high input impedance (1020 Ω)\n\nwas used to ensure that the measurement of the guard ring voltage does not influence the\n\nsensor and modify the results. As guard rings are floating structure unconnected to any\n\nbias source, using a standard voltmeter with lower impedance would draw current from\n\nthe guard ring modifying its electrical potential . N-in-p sensor were simulated using an\n\nn-bulk as it has been observed in the last section that the bulk type has been inverted by\n\nthermal donors.\n\nThe first measurements were performed on the three guard ring structure presented in\n\nfigure 3.9. Figure 4.16 show the simulated and measured guard ring potential for the three\n\nguard ring models. The results obtained are in good agreement with the experimental data,\n\ngiving us confidence that the TCAD simulation model correctly represent the physical be-\n\n136\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\n(a) N implant\n\n(b) P implant\n\nFigure 4.15 – Comparison of simulated and measured n and p doping profile of the\nPPSU09 production after optimization of implantation and diffusion parameters\n\n137\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\n(a) ATLAS standard structure\n\n(b) Small guard ring structure\n\n(c) Large guard ring structure\n\nFigure 4.16 – Comparison of simulated and measured guard ring potential of the PPSU09\nproduction guard ring structure after optimization of implantation and diffusion parameters\n\n138\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nhavior of multi-guard ring structures. Some discrepancies that could not be easily explained\n\ncan be seen in the outer guard rings of the small guard ring n-in-p structure. The mea-\n\nsurement of these guard rings have shown poor reproducibility hinting at some transient\n\nphenomenon. A possible explanation could be an oscillation of the guard rings around the\n\nedge potential due to some unknown effect that wasn’t reproduced in simulation. The use\n\nof the experimental doping profile data has been very important to obtain correct results.\n\nIt has been observed that the guard ring’s bias potential obtained in simulation was very\n\ndependent on the doping profile characteristics of the n, p and p-spray implants. The use\n\nof the profile measurement allowed us to reduce the number of free parameters needed for\n\nthe simulation and resulted in the correct guard ring behavior.\n\nThe size of the guard rings of the three model studied here doesn’t allow easy measure-\n\nments. Contact with the guard ring has to be done by etching mechanically the passivation\n\nlayer over the guard ring structure using the probe station needle, which can eventually\n\nreduce the performance of the sensor. A limited number of full guard ring structure were\n\nmeasured to keep its integrity for future use for irradiation or beam test studies. As men-\n\ntioned in the first section of this chapter, a set of test diodes with large contactable guard\n\nrings was placed in both the PPSU09 n-in-n and n-in-p production. The number of guard\n\nring on the diode span from 1 to 4, the same guard ring size for each structure. For example,\n\nthe 4 guard ring model contained the 3 guard rings of the 3 guard ring diode plus a fourth\n\none added on the outer edge of the structure. The implant of the guard ring structure are\n\n60 µm wide, with the electrode width being respectively, from the inner to outer guard\n\nring, 120, 150,180 and 210 µm. The electrode overshoot the implant toward the outer edge\n\nof the diode by 18 µm and the distance between each ring is 10 µm. Measurement of the\n\nguard ring potential of the structure was done for the diodes of the PPSU09 n and p bulk\n\nproduction. Measurements within each production have shown a variation of less than 10\n\n% between the measured guard ring potential for the diodes coming from different wa-\n\nfer. A good reproducibility has been observed as the same structure have led to the same\n\nresults when measured two months after the first measurements. Two set of four diodes\n\nn-in-n diodes have been irradiated at CERN with 24 GeV protons to 1× 1015neqcm\n−1 and\n\n4×1015neqcm\n−1 . A set of n-in-p diode was also irradiated to a fluence of 2×1015neqcm\n\n−1.\n\n139\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nThe sample were kept under 0 C after irradiation during transport and were kept at -28C\n\nduring storage to avoid annealing effects.\n\nA TCAD simulation of the four diodes was performed for each type of bulk, before and\n\nafter irradiation. Figure 4.17 show the comparison of the simulated and measured guard\n\nring potential for four diodes of the n-in-p production. The removal of the outer guard\n\nrings made no change to the guard ring potential of the inner guard rings and only results\n\nof the 4 guard ring simulation are shown in this figure. This is analog to the simulation\n\nof the removal of the ATLAS standard guard ring structure similar to what described in\n\nchapter 3. In both case, the removal of the outer guard ring have only minor influence\n\non the potential taken by the other guard rings. Simulation reproduce well the guard ring\n\npotential curves at lower bias voltage but fails at higher voltage. However, the general\n\nbehavior of the guard ring structure is well reproduced by the simulation. Other guard\n\nring measurements results have shown better agreement with simulation The discrepancy\n\nbetween simulation and measurements could be induced by a physical phenomenon not\n\nincluded in the simulation or systematic effect in the measurements when measuring large\n\nguard ring as it is the case here.\n\nThe n-in-n diode simulation have shown the same results as for the n-in-p diode si-\n\nmulation. The guard ring behavior seems to be independent of the substrate type. The\n\nguard ring structure have been measured on the n-in-p and n-in-n irradiated diodes. Fi-\n\ngure 4.18 show the measured potential on 2 of the diodes irradiated to a dose of less than\n\n2×1015neqcm\n−1]. The guard ring seems, in both cases , to completely stop working. In the\n\ncase of the n-in-n structure, this was predicted to happen after irradiation. For the n-in-p\n\nstructure, the simulation predicts working guard rings. However, it was discovered that\n\nincreasing the surface charge up to a value of 2× 1012cm2 produce the observed effect. In\n\nthe simulation, the guard ring stay shorted to ground potential for any bias voltage. The\n\nvariation seen in the experimental data are not reproduced in details but the very low value\n\nand the uniformity of the potential on the guard rings show that they are not functional\n\nany more and the value measured is probably the result of a systematic effect present in\n\nthe measurement setup.\n\nThe p-spray insulation used and measured in the n-in-p production might not be suffi-\n\n140\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nFigure 4.17 – Comparison of measurement and TCAD simulation of the potential of the\nLAL diode test structure for the n-in-p production\n\ncient to ensure proper insulation of the guard rings after irradiation with charged particles.\n\nNeutron irradiation of these diode structure should be performed to evaluate the decouple\n\nthe measurement and pinpoint the different effect affecting the guard ring. P-spray si-\n\nmulation seems to overestimate the insulation power of the implant and further work is\n\nneeded.\n\n4.1.3 Current versus Bias characteristics\n\nAn important parameter of pixel sensor are its leakage current and breakdown voltage.\n\nSimulation of leakage current represent a difficult task as all simulation are performed in\n\ntwo dimension, on sub-volume of the actual detector. Breakdown voltage prediction can\n\nhowever be done from simulation and compared with experimental results. Leakage current\n\nwas measured in the various FE-I3 and FE-I4 structure of the n-in-n and n-in-p produc-\n\ntion. Figure 4.19 show the results of these measurements for the FE-I3 sensors. The n-in-p\n\nproduction contained two types of sensor using respectively the p-spray and the modera-\n\nted p-spray insulation method. It can be clearly observed that p-spray sensors have lower\n\n141\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\n(a) n-in-n,1× 1015neqcm\n−1\n\n(b) n-in-p,2× 1015neqcm\n−1\n\nFigure 4.18 – Measured guard ring potential for LAL diodes after irradiation\n\n142\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nbreakdown voltage, between 200 and 250V, as the p-spray implants surrounding the n im-\n\nplants generate high electric field region where high leakage current can be generated. The\n\nbreakdown observed in these sensor is not a steep brutal one but rather a slow exponen-\n\ntially raising leakage current. The moderated p-spray sensors show breakdown around 400\n\nV. In moderated p-spray, the p implant surrounding the n implants is less deep than the\n\np-spray implant and this has for effect of reducing the electric field and leakage current\n\nin these regions. Simulation predict a much higher breakdown voltage for these sensors,\n\naround 1000V. It is however known that other sensor of the same production show higher\n\nbreakdown voltage. The wafer used for these measurement was discarded and considered\n\nas a bad wafer in view of its lower breakdown voltages. The wafer is however useful for\n\nsensor characterization and irradiation studies. FE-I4 sensors with p-spray and moderated\n\np-spray were also present on this wafer and leakage current measurement are shown in\n\nfigure 4.20. Again in this case the moderated model show better breakdown voltage but\n\nstatistics was too low to draw any conclusion.\n\nThe guard ring diode were also extensively measured before and after irradiation. At\n\nthe moment of writing these line, no cooled probe station chuck were available to perform\n\nmeasurements on the irradiated diodes at a lower temperature to limit leakage current.\n\nThe results shown here are of limited interest but still show the increase of leakage current\n\nwith exposed fluence. Figure 4.21 show the IV characteristics of the diodes that were\n\navailable for measurements. An interesting behavior can be observed in the case of the n-\n\nin-p diodes. The leakage current curves shown here for these sensors were taken successively\n\non the same diode during overnight runs. The breakdown voltage observed in the curves\n\nfor the n-in-p diodes drifted during the night getting higher until it settles to a stable\n\nhigher value. It can be interpreted as a reorganization of the surface charge within the\n\noxide layer due to the added presence of an electric field. Local hotspot breakdown due to\n\nlocally accumulated charge are diluted as the charge drift in the field present in the oxide,\n\nrevealing the real breakdown value of the diode. The big variation in breakdown from a\n\ndiode to another, does not allow to draw conclusion on the relation between the number of\n\nguard ring and the breakdown voltage of the diodes. The n-in-n diode measurement show\n\nthe only measurement available for n-in-n diode at the moment of writing these line. Most\n\n143\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\n(a) n-in-n production\n\n(b) n-in-p production\n\nFigure 4.19 – Leakage current in the FEI3 of the PPSU09 production\n\n144\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nFigure 4.20 – n-in-p small guard rings FE-I4 leakage current\n\ndiode of the production were sent to irradiation for a short time period.\n\nFigure 4.22 show the leakage current measured in the diode after irradiation. As measu-\n\nrement were taken at room temperature, very high leakage current value can be observed.\n\nMeasurement were performed very quickly to avoid the heating of the diode that was also\n\npromptly put back to its cold storage. No apparent breakdown can be observed in any of\n\nthe diode. A slow rise of current, also observed in the current-voltage curve simulated in\n\nchapter 3. The breakdown present before irradiation are quenched by the presence of the\n\nhigh amount of defects trapping the charge before it can multiply and create a cascade\n\nleaking to breakdown. Leakage value for the different diodes irradiated at the same fluence\n\nis uniform showing that leakage current after irradiation is dominated by bulk defect ge-\n\nneration recombination current and not surface quality and edge effects as usually seen\n\nbefore irradiation.\n\nExperimental measurements presented in this section were used to better understand\n\nthe behavior of the sensors and interpretation of the simulation results can be used to\n\ngain understanding of the experimental data. Guard ring potential are well reproduced\n\n145\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\n(a) n-in-n production\n\n(b) n-in-p production\n\nFigure 4.21 – Leakage current in the guard ring diodes of the PPSU09 production\n\n146\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\n(a) n-in-n production\n\n(b) n-in-p production\n\nFigure 4.22 – Leakage current in the guard ring diodes of the PPSU09 production after\nirradiation\n\n147\n\n\n\n4.1. EXPERIMENTAL VALIDATION OF TCAD SIMULATION\n\nby simulation, both before and after irradiation, in the case studied here, and lead us\n\nto be confident in the results coming from TCAD simulation. The use of experimental\n\ndoping profile allow easy plug-and-play reproduction of the guard ring structure behavior.\n\nThe model presented in this work can be used for optimization of guard ring structure for\n\nfuture sensor such as the super LHC pixel detector. Leakage current and breakdown voltage\n\nremain hard to predict but the soft breakdown observed in radiation damage simulation is\n\nalso observed in irradiated devices.The breakdown voltage value predicted by the simulation\n\nseems to be higher than the one observed in reality. The qualitative tendency can however\n\nbe used to gain insight on the effect of modifications to the design of multi-guard ring\n\nstructure.\n\n148\n\n\n\nChapitre 5\n\nPlanar Pixel Sensor digitization for\nATLAS IBL simulation\n\nThe ATLAS IBL pixel sensor will be subject to extensive radiation damage that will\n\naffect its performance in terms of reconstruction of the charged particles trajectory and\n\nimpulsion. Charge collection efficiency will be reduced by trapping and high electric field\n\nwill be present in the bulk of the sensor. Space charge inversion will eventually occur\n\nand sensor will futher need to be operated underdepleted as bias voltage sufficient for full\n\ndepletion will not be reachable by the power supply of the detector system. To produce\n\naccurate simulation of the full detector system and evaluate reconstruction performances,\n\nwe need an accurate and fast model of the charge collection to digitize the energy deposition\n\ninformation coming from the GEANT4 simulation of the detector. Our group has developed\n\na digitizer based on our knowledge of TCAD and Monte-Carlo simulation of pixel detectors.\n\nA simulation framework , the ALLPix software, was developed. It provides an easy\n\ntest-bench for digitizer using the GEANT4 [51] simulation package. The software can be\n\nused to simulate any pixel sensor geometry along with its surrounding environment. A\n\nrealistic model of the EUDET pixel telescope [52], shown in figure 5.1, was used in the\n\ntest beam period of November 2009, july and november 2010 at CERN SPS (120 GeV\n\npions) and in May 2010 at DESY (1-4 GeV electrons). It provides a simulation tool of\n\nthe telescope that is useful for debugging and understanding the behavior of the telescope.\n\nVarious digitization model can be used to convert the GEANT4 hits of the device under\n\ntest into detector hit information. The simulation can be used to generate virtual telescope\n\n149\n\n\n\ndata that can be reconstructed using the telescope reconstruction software. The comparison\n\nbetween the real test beam data and simulated one will be used to validate the digitizer\n\npackage developed at LAL. Once validated. this digitizer can then be transferred to the\n\nATLAS simulation software to perform full detector simulation.\n\nFigure 5.1 – EUDET telescope geometry implemented in the ALLPix framework. Wire\nframe box represent the 6 telescope planes and the green volumes in the middle are the\nDUT PCB card\n\nThe LAL digitization model presented here aims to provide a simple digitization tool\n\nfor the FE-I3 and FE-I4 planar pixel sensor featuring the slim edges guard ring structures\n\nproposed for the IBL pixel detector. The radiation damage model used to compute the\n\ntrapping time and resistivity of the bulk after exposure to radiation is the Hamburg model\n\npresented in chapter 2.\n\nTo simulate charge transport in the bulk of the pixel sensor, we first take the trajectory\n\nof the particle crossing the sensitive volume and divide it into a smaller fraction of charge\n\n150\n\n\n\ndeposition hits. The energy deposition of each hit is translated in a number of electron hole\n\npair and variation of this number following the Poisson statistics and the Fano factor is\n\ntaken into account. The number of these hit can be set as a free parameter to allow more or\n\nless accurate representation of the charge deposition along the track. The punctual charge\n\nelement are then transported in the electric following the drift-diffusion equation. Effects\n\nsuch as saturation velocity are taken into account to accurately compute the drift time of\n\nthe punctual charge element. Electric field can be provided through a TCAD simulation\n\nresult of evaluated using an analytical function. The integration of the trajectory is done\n\nusing a fifth order integration method called Runge-Kutta-Fehlberg method [53]. For most\n\nsimulation purpose, the drift inside unidimensional electric field should be sufficient to\n\nreproduce well the experimental data. A schematic of the digitization charge transport\n\nmodel can be seen in figure 5.2, where the drift cone of the charge element are shown\n\nspreading from the track position along the drift trajectory.\n\nFigure 5.2 – Schematic of the digitization model\n\nA lifetime is also computed for each charge element hole and electrons using the trapping\n\nlifetime computed with the Hamburg model. The charge element trapping position in the\n\nsensor is determined using the drift trajectory and velocity of each charge element. Using\n\nthe initial and final position of the charge element, we can use Ramo theorem (equation\n\nThe lateral diffusion associated to each charge element is also computed using the\n\ndrift time of the charge element and the classical solution of the diffusion equation for a\n\n151\n\n\n\n5.1. TEST BEAM VALIDATION OF TCAD SIMULATION AND DIGITIZATION\n\ngaussian distribution. Equation 5.1 describe the evolution of the standard deviation of the\n\ndistribution with time in silicon. D is the diffusion coefficient.\n\nσx,y,z(t) =\n√\n\n2De,ht (5.1)\n\nThe resulting gaussian distribution of charge is then projected on the pixel plane, and\n\ncharge inside the element is divided between the pixels using equation 5.2\n\nQi =\n\n∫ xf i\n\nx0i\n\n∫ yf i\n\ny0i\nρ(x, y)dxdy (5.2)\n\nwhere Qi is the charge collected by electrode i, bounded by the rectangle extending\n\nfrom x0i to Xf i and y0i to yf i and ρ(x, y) the projected gaussian distribution of the\n\ncharge element on the pixel surface. Finally, the threshold of the readout chip threshold\n\nis simulated by eliminating pixel hit under a constant threshold that is a free parameter\n\nof the model. This model was implemented in a digitizer for the allpix framework. Results\n\nand comparison with test beam data will be shown in the next section.\n\n5.1 Test beam validation of TCAD simulation and digitiza-\ntion\n\nThe EUDET telescope has been used to study the tracking performances of the different\n\nflavor of ATLAS pixel sensor from the PPSU production. The telescope is composed of 6\n\nplanes of MIMOSA26 MAPS sensor with a pitch of 18.5 µm arranged in a matrix of\n\n1156x1156 pixels. The telescope is divided in two arms located each side of the ATLAS pixel\n\ndetector assemblies. Particles crossing the entire telescope were selected using 4 scintillator\n\ntriggers in coincidence two by two on each end of the telescope assembly. Data taken by\n\nthe ATLAS pixel assemblies are recorded for the 16 next level 1 trigger count. The EUDET\n\nsoftware is then used to reconstruct the trajectories of the recorded particles. The tracks\n\nare extrapolated to the device under test and used to analyze the behavior of the ATLAS\n\npixel sensor. Track positioning resolution between the telescope arms is 3 µm.\n\n152\n\n\n\n5.1. TEST BEAM VALIDATION OF TCAD SIMULATION AND DIGITIZATION\n\n5.1.1 Validation of the digitization model\n\nI will present here a validation of the digitizer using the reconstructed data from CERN\n\nplanar pixel October 2010 test beam period. The devices under test that were evaluated\n\nwere , in increasing order distance from the beam : a 11 guard ring ATLAS n-in-n sensor\n\nfrom the PPSU09 production, a n-in-n stepwise shifted pixel structure and a n-in-p unir-\n\nradiated standard small guard ring pixel detector also from the PPSU09. A fourth sensor\n\nfrom Hammamatsu was present in the beam but it not considered in this analysis. The data\n\ncontained 6.89 million events taken from run 20275 to 20358. Bias voltage were maintained\n\nat 150 V through the run and sensor were operated at room temperature. A simulation of\n\n500 000 trigger event were accumulated using the telescope geometry to compare with the\n\ndata from the pixel sensor under test.\n\nSimulation was performed for a 250 µm thick 5000 kΩcm−1 sensor biased at 150 V.\n\nElectric field was considered from a TCAD simulation of a single pixel cell for a cutline\n\ntaken at its middle through the depth of the model. The diffusion coefficient for electrons\n\nwas 40.2 cm2/s and mobility was computed using the field dependent model shown in\n\nchapter 2.\n\nFigure 5.3 and 5.4 show the unbiased residual of the track position reconstructed with\n\nthe telescope with respect to the reconstructed hit position in the devices under test.\n\nThree methods of hit position reconstruction are used and compared in the test beam data\n\nreconstruction. The digital and analog method consist respectively of computing the center\n\nof gravity of the hit pixel that are regrouped in a cluster and calculating the charge-weighted\n\ncenter of gravity of the cluster’s pixel. The simulation data were reconstructed in the same\n\nmanner and figure ?? and ?? show the residual distribution between the Monte-Carlo truth\n\nparticle hit position and the reconstructed position from digitization data.\n\nCluster size distribution of the device under test can be seen in figure 5.6 along with the\n\nsimulated data. The simulation reproduce well within 3.5% the size distribution of clusters\n\nin test beam data for all three sensors. The Time-Over-Threshold (TOT) distribution in\n\nthe pixel X direction can be seen in figure ?? for the three devices tested and for the\n\nsimulation of the devices along with the results from simulation.\n\n153\n\n\n\n5.1. TEST BEAM VALIDATION OF TCAD SIMULATION AND DIGITIZATION\n\n(a) X direction (400 µm pitch)\n\n(b) Y direction (50 µm pitch)\n\nFigure 5.3 – Experimental unbiased residual distribution for the planar sensors between\nthe arms of the telescope, for cluster size 1\n\n154\n\n\n\n5.1. TEST BEAM VALIDATION OF TCAD SIMULATION AND DIGITIZATION\n\n(a) X, charge weighted mean (b) X, geometrical mean (c) X, Max TOT\n\n(d) Y, charge weighted mean (e) Y, geometrical mean (f) Y, Max TOT\n\nFigure 5.4 – Experimental unbiased residual distribution for the planar sensors between\nthe arms of the telescope, for cluster size 2\n\n(a) X, charge weighted mean (b) X, geometrical mean (c) X,Max TOT\n\n(d) Y, charge weighted mean (e) Y, geometrical mean (f) y Max TOT\n\nFigure 5.5 – Simulated unbiased residual distribution for the planar sensors between the\narms of the telescope\n\n155\n\n\n\n5.1. TEST BEAM VALIDATION OF TCAD SIMULATION AND DIGITIZATION\n\nFigure 5.6 – Cluster size distribution in experimental Pion test beam data compared to\ndigitization\n\n(a) 15 GR n-in-n (b) stepwise structure\n\n(c) n-in-p small guard rings (d) Simulation\n\nFigure 5.7 – TOT Profile in the X pixel direction, average of all pixels\n\n156\n\n\n\n5.1. TEST BEAM VALIDATION OF TCAD SIMULATION AND DIGITIZATION\n\nThe simulation results are in good agreement with the results obtained with the three\n\ndevice in test beam. The Charge profile along the X direction are different for the expe-\n\nrimental dara compared to simulated data because of the punch-trough structure, seen in\n\nfigure 5.8, which is used to bias the pixel and collect charges so that it does not reach\n\nthe readout chip. This mechanism is not included in the simulation. The TOT average\n\nvalue from simulation is also lower than in the experimental data because the sensor was\n\nsimulated with a thinner bulk. The calibration of the virtual chip was kept to obtain a\n\nvalue of 60 for a deposited charge of 22000 electrons.\n\nFigure 5.8 – FE-I3 standard pixel with its punch-trough bias structure\n\n5.1.2 Edge effects\n\nThe stepwise ATLAS pixel structure has been designed to study the edge effects of the\n\nslim edge structure and verify the predictions of the TCAD simulation of the structure.\n\nFigure 5.9 show the geometry of edge pixel that was used in the production. Edge pixels are\n\nshifted by groups of 8 by steps of 25 microns under the guard ring structure. The efficiency\n\nand charge collection averaged over each set of 8 pixels can be seen in figure 5.10.\n\nFigure 5.9 – GDS drawing of the stepwise pixel structure\n\nThe charge collected under the guard rings is reduced with respect to the charge collec-\n\nted within the guard ring structure. This effect was predicted from TCAD simulation and\n\n157\n\n\n\n5.1. TEST BEAM VALIDATION OF TCAD SIMULATION AND DIGITIZATION\n\n(a) Detection efficiency at the edge of the sensor\n\n(b) Charge collection profile along a pixel shifted by 200\n\n(c) Charge collection profile along the shifted pixels\n\nFigure 5.10 – Experimental measurement of the charge collection and detection efficiency\nof the slim edge guard ring structure\n\n158\n\n\n\n5.1. TEST BEAM VALIDATION OF TCAD SIMULATION AND DIGITIZATION\n\ndemonstrated in test beam situation . The detection efficiency is maintained as signal is\n\nlarge enough to trigger the readout electronics. Post-irradiation behavior, with guard rings\n\nloosing their usefulness and SCSI occurring, the difference in charge collection in the active\n\nzone and under the guard ring should be minimal. These data can be used to parametrize\n\nthe charge collection at the edge pixels in the digitizer to simulate the slim edge structure.\n\nOverall , the digitization model reproduce well the behavior of ATLAS FE-I3 unir-\n\nradiated and irradiated sensor tested at CERN using the EUDET telescope. This model\n\ncan be scaled to simulate FE-I4 data and could be useful for future beam test using pixel\n\nsensors. The digitizer that was developed can be used in the general ATLAS simulation\n\nsoftware to take into account effects such parameters like thickness of the sensor and pixel\n\npitch size. This will be used in the IBL ATLAS detector simulation software to simulate\n\nthe slim edge planar sensor candidate for its sensor. The possibility to use TCAD electric\n\nfield and Ramo Potential , including the mechanism of trapping in the simulation is an\n\nideal tool for a detailed simulation of irradiated sensors and should be used to evaluate the\n\nperformances of the ATLAS pixel detector after irradiation .\n\n159\n\n\n\n5.1. TEST BEAM VALIDATION OF TCAD SIMULATION AND DIGITIZATION\n\n160\n\n\n\nConclusion\n\n161\n\n\n\n\n\nCONCLUSION\n\nTCAD simulation have been used for a long time in various engineering field to study\n\ncomplex system and help with their design. The fast computer available today are now\n\nallowing the simulation of large structure using complex model to represent the physics\n\nimplicated in the operation of planar pixel sensor and other semiconductor detection de-\n\nvices. In this work, I presented various simulation results on planar pixel sensor under\n\nvarious condition of operation. It has been shown that reduction of inactive area in pla-\n\nnar pixel sensors can be achieve through the reduction of the number of guard rings and\n\nby using the slim edge structure. The thinning of sensors have been shown to be a valid\n\nmethod to obtain a depleted sensor at lower bias voltage while maintaining comparable\n\nbreakdown voltage associated to thick sensor featuring‘ the same number of guard rings.\n\nCutting edge width reduction have been shown to not influence the lateral depletion of the\n\nsensor. Finally, anomalous charge collection observed in highly irradiated silicon sensor has\n\nbeen explained to first principles using TCAD simulation\n\nThe comparison between experimental data and TCAD simulation has demonstrated\n\nthat TCAD simulation can lead to quantitative results when tuned with the correct process\n\nparameters using the adequate physical models. Further work will be needed to obtain more\n\nquantitative results concerning breakdown voltage for unirradiated and irradiated silicon\n\nsensors. Qualitatively, results obtained here are however encouraging and further work\n\nwill certainly lead to more reasonable results in the near future. Behavior of irradiated\n\nsensors has been reproduced in many aspects, from charge collection and change in bulk\n\nresistivity to double junction formation and failure of the guard ring structures. Further\n\nirradiation studies with test structures will be needed to improve the prediction power of\n\nTCAD simulation. Lower fluences should be explored to observe the failure point of guard\n\nring structure.\n\nBeam test activities have shown that planar pixel sensor can be operated with reduced\n\ninactive area and deliver the same performance as the actual planar pixel detector used in\n\nATLAS inner detector. The need to study fine effects present in these device brought us to\n\ndevelop a simulation package to study the digitization models proposed for our device and\n\ncompare easily with beam test data taken at CERN SPS and DESY Electron Synchrotron.\n\nA digitization model using the Ramo potential and TCAD electric field simulation was\n\n163\n\n\n\nCONCLUSION\n\ndeveloped to help with the study of the new sensor prototypes and to be used in full scale\n\nATLAS and IBL simulation. Comparison with beam test data show the model reproduce\n\nwell the data for unirradiated sensors with various geometry. The model proposed can\n\nbe used to simulate trapping and thinning effects in irradiated sensors. Comparison with\n\nirradiated data and with FEI4 readout should be done to validate further the digitization\n\nmodel.\n\nThe future of silicon detector will be exciting. The arrival of 3D electronics, which\n\nallow to build four-side buttable device and readout electronics using different process for\n\ndifferent functions ,which I had the privilege to see during my thesis, will change our way\n\nto work with silicon sensors and will allow for faster, more radiation hard detector that\n\nexhibit less inactive zones. The first 3D electronic ASIC, the OMEGAPIX, shown in figure\n\n5.11, was received at LAL and open a new era for HEP experiment. I hope my contribution\n\nto the design of the omegapix-2 will lead to exciting new ASICs for the next generation of\n\npixel sensors.\n\nFigure 5.11 – The omegapix analog and digital tier\n\nPersonally ... Leaving your country, your family and your friends to pursue a career\n\non an other continent is something I never thought I would do. It is a very difficult choice\n\n164\n\n\n\nCONCLUSION\n\nto make but it is one I do not regret. During the last three year and half, I had the pleasure\n\nto work in one of the most amazing laboratory I have been honored to know. I travelled\n\naround the world to such exotic places as Alger, Bumerovska or Knoxville. I was granted a\n\nlot of autonomy and for that I thanks all the people who made this possible. I am not the\n\nsame person today I was when I first left home with my package on my back. I discovered\n\nthe real pleasure there can be in making errors and finding the good way to do things by\n\nmyself. There was a lot of frustration, a huge amount of work that do not always show up in\n\nthis thesis because choice were made and sometime because this work is hard to valorize.\n\nOne of my greatest achievement during this thesis, in my opinion, was to build a clean\n\nroom setup that is competitive and can reproduce results taken in other labs where clean\n\nroom activities have been going on for many years, Unfortunately, sometime experimental\n\nwork does not go as fast as you would like it to go. I hope the students who will take\n\nmy place in this group will appreciate the work behind these setup, which will be working\n\nfrom day one for them, and will make good use of it to further advance the accuracy of\n\nour simulation models.\n\nThe knowledge I accumulated here will certainly follow me in my career and I hope to\n\nbe able to continue it in the exciting field of experimental high energy physics as long as\n\npossible.\n\n165\n\n\n\nCONCLUSION\n\n166\n\n\n\nWord Cloud\n\n167\n\n\n\n\n\nWORD CLOUD\n\n169\n\n\n\nWORD CLOUD\n\n170\n\n\n\nBibliographie\n\n[1] O. S. Brüning, P. Collier, P. Lebrun, S. nd Myers, R. Ostojic, J. Poole, and P. Proud-\n\nlock, LHC Design Report. Geneva : CERN, 2004. 17\n\n[2] G. Aad and al., “Atlas pixel detector electronics and sensors,” J. Isnt., vol. 3, p. 7007,\n\n2008. 20, 21\n\n[3] S. Coellia and the ATLAS Pixel Collaboration, “Mechanics and infrastructure for the\n\natlas pixel detector,” J. 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Fehlberg, “Low-order classical runge-kutta formulas with stepsize control,” NASA\n\nTechnical Report, pp. R–315, 1969. 151\n\n\n\n\n\tIntroduction\n\tThe ATLAS experiment and upgrade project \n\tThe Large Hadron Collider\n\tThe ATLAS experiment\n\tThe Inner detector\n\tThe calorimeter\n\tThe muon spectrometer\n\n\tTHe ATLAS upgrade projects\n\tPhase 1 : The Insertable B-Layer (IBL)\n\tPhase 2 : Upgrade for high luminosity\n\n\n\tPrinciples of Silicon pixel sensors \n\tThe physics of Silicon\n\tSemiconductors properties\n\tCharge transport\n\tThe pn junction\n\tPhysical models\n\n\tRadiation detection\n\tThe energy deposition process\n\tSignal formation\n\n\tThe Hybrid Planar Pixel Sensor\n\tOther Silicon sensors\n\tthe 3D pixel sensor\n\tHigh Resistivity Monolithic Active Pixel Sensors (MAPS)\n\n\tRadiation damage in Silicon sensors\n\tNon-ionizing Energy Loss (NIEL)\n\tIonizing energy loss\n\n\n\tTCAD Simulation models \n\tProcess simulation\n\tDevice simulation\n\tGeometry\n\tboundary conditions\n\n\tThe Multi-Guard Ring structure\n\tPrinciples of guard ring structures\n\tOptimization of guard ring structures for reduction of inactive area and radiation hardness \n\tThe Slim Edge Guard Ring structure\n\n\tThe charge amplification mechanism in highly irradiated silicon sensors\n\n\tFrom TCAD simulation to experimental data\n\tExperimental validation of TCAD simulation\n\tDoping profile measurements\n\tGuard Ring measurements\n\tCurrent versus Bias characteristics\n\n\n\tPlanar Pixel Sensor digitization for ATLAS IBL simulation\n\tTest beam validation of TCAD simulation and digitization\n\tValidation of the digitization model\n\tEdge effects\n\n\n\tConclusion\n\tWord Cloud\n\tBibliography"},"fulltext":true,"key":"c56fd24ba5eaaa333c2759245c075b77","url":"https://inspirehep.net/files/c56fd24ba5eaaa333c2759245c075b77"},{"filename":"VA_BENOIT_MATHIEU_10062011.pdf","key":"ee6767ebdc120b7c440d480d25024424","url":"https://inspirehep.net/files/ee6767ebdc120b7c440d480d25024424"}],"citation_count_without_self_citations":1,"report_numbers":[{"value":"LAL-11-118"},{"value":"tel-00610015"},{"value":"2011PA112070"}],"authors":[{"raw_affiliations":[{"value":"Laboratoire de l'Accélérateur Linéaire, France"}],"full_name_unicode_normalized":"benoit, mathieu","full_name":"Benoit, Mathieu","record":{"$ref":"https://inspirehep.net/api/authors/1070425"},"last_name":"Benoit","ids":[{"schema":"INSPIRE ID","value":"INSPIRE-00211411"},{"schema":"INSPIRE BAI","value":"M.Benoit.1"}],"affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/903100"},"value":"Orsay, LAL"}],"signature_block":"BANATm","uuid":"7520994c-e285-4734-b967-0cc7db55b701","first_name":"Mathieu","recid":1070425}],"citation_count":1,"$schema":"https://inspirehep.net/schemas/records/hep.json","keywords":[{"source":"author","value":"Radiation Damage"},{"source":"author","value":"IBL"},{"source":"author","value":"SLHC"},{"source":"author","value":"Test Beam"},{"source":"author","value":"Slim edges"},{"source":"author","value":"Planar Pixel Sensor"},{"source":"author","value":"Digitization"},{"source":"author","value":"TCAD simulation"},{"source":"author","value":"Silicon detector"},{"source":"author","value":"-Guard Ring Structure"},{"source":"author","value":"Dommage induit par la radiation"},{"source":"author","value":"Silicium"},{"source":"author","value":"Détecteur pixel planaires"},{"source":"author","value":"Simulation TCAD"},{"source":"author","value":"Test faisceau"},{"source":"author","value":"Numérisation"},{"source":"author","value":"SLHC;"},{"schema":"INSPIRE","value":"thesis"},{"schema":"INSPIRE","value":"semiconductor detector: pixel"},{"schema":"INSPIRE","value":"radiation: damage"},{"schema":"INSPIRE","value":"charge: yield"},{"schema":"INSPIRE","value":"detector: design"},{"schema":"INSPIRE","value":"model"},{"schema":"INSPIRE","value":"ATLAS"},{"schema":"INSPIRE","value":"numerical calculations"}],"references":[{"reference":{"label":"3","misc":["The,. 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Different physical models available have been studied to develop a coherent model of radiation damage in silicon that can be used to predict silicon pixel sensor behavior after exposure to radiation. The Multi-Guard Ring Structure,a protection structure used in pixel sensor design was studied to obtain guidelines for the reduction of inactive edges detrimental to detector operation while keeping a good sensor behavior through its lifetime in the ATLAS detector. A campaign of measurement of the sensor's process parameters and electrical behavior to validate and calibrate the TCAD simulation models and results are also presented. A model for diode charge collection in highly irradiated environment was developed to explain the high charge collection observed in highly irradiated devices. A simple planar pixel sensor digitization model to be used in test beam and full detector system is detailed. It allows for easy comparison between experimental data and prediction by the various radiation damage models available. The digitizer has been validated using test beam data for unirradiated sensors and can be used to produce the first full scale simulation of the ATLAS detector with the IBL that include sensor effects such as slim edge and thinning of the sensor."},{"source":"TEL","value":"In this work, is presented a study, using TCAD simulation, of the possible methods of designing of a planar pixel sensors by reducing their inactive area and improving their radiation hardness for use in the Insertable B-Layer (IBL) project and for SLHC upgrade phase for the ATLAS experiment. Different physical models available have been studied to develop a coherent model of radiation damage in silicon that can be used to predict silicon pixel sensor behavior after exposure to radiation. The Multi-Guard Ring Structure,a protection structure used in pixel sensor design was studied to obtain guidelines for the reduction of inactive edges detrimental to detector operation while keeping a good sensor behavior through its lifetime in the ATLAS detector. A campaign of measurement of the sensor’s process parameters and electrical behavior to validate and calibrate the TCAD simulation models and results are also presented. A model for diode charge collection in highly irradiated environment was developed to explain the high charge collection observed in highly irradiated devices. A simple planar pixel sensor digitization model to be used in test beam and full detector system is detailed. It allows for easy comparison between experimental data and prediction by the various radiation damage models available. The digitizer has been validated using test beam data for unirradiated sensors and can be used to produce the first full scale simulation of the ATLAS detector with the IBL that include sensor effects such as slim edge and thinning of the sensor.","abstract_source_suggest":{"input":"TEL"}},{"source":"dart-europe.org","value":"Le Large Hadron Collider (LHC), située au CERN, Genève, produit des collisions de protons accélérés à une énergie de 3.5 TeV depuis le 23 Novembre 2009. L’expérience ATLAS enregistre depuis des données et poursuit sa recherche de nouvelle physique à travers l’analyse de la cinématique des événements issues des collisions. L’augmentation prévue de la luminosité sur la période s’étalant de 2011 2020 apportera de nouveaux défis pour le détecteur qui doivent être considérés pour maintenir les bonnes performance de la configuration actuelle. Le détecteur interne sera le sous-détecteur le plus affecté par l’augmentation de la luminosité qui se traduira par une augmentation des dommages occasionnés par la forte radiation et par la multiplication du nombre de traces associées à chaque croisement de faisceau. Les dommages causés par l’irradiation intense entrainera une perte d’efficacité de détection et une réduction du nombre de canaux actifs. Un intense effort de Recherche et Développement (R&D) est présentement en cours pour concevoir un nouveau détecteur pixel plus tolérant aux radiations et au cumul des événements générant un grand nombre de traces à reconstruire. Un premier projet de mise-à-jour du détecteur interne, nommé Insertable B-Layer (IBL) consiste à ajouter un couche de détection entre le tube à vide du faisceau et la première couche de silicium. Le projet SLHC prévoit de remplacer l’ensemble du détecteur interne par une version améliorée plus tolérante aux radiations et aux cumuls des événements. Dans cet ouvrage, je présente une étude utilisant la simulation technologique assisté par ordinateur (TCAD) portant sur les méthodes de conception des détecteurs pixels planaires permettant de réduire les zones inactives des détecteurs et d’augmenter leurs tolérances aux radiations. Les différents modèles physiques disponible ont étés étudiés pour développer un modèle cohérent capablede prédire le fonctionnement des détecteurs pixels planaires après irradiation. La structure d’anneaux de gardes utilisée dans le détecteur interne actuel a été étudié pour obtenir de l’information sur les possible méthodes permettant de réduire l’étendu de la surface occupée par cette structure tout en conservant un fonctionnement stable tout au long de la vie du détecteur dans l’expérience ATLAS. Une campagne de mesures sur des structures pixels fut organisée pour comparer les résultats obtenue grâce à la simulation avec le comportement des structures réelles. Les paramètres de fabrication ainsi que le comportement électrique ont été mesurés et comparés aux simulations pour valider et calibrer le modèle de simulation TCAD. Un modèle a été développé pour expliquer la collection de charge excessive observée dans les détecteurs planaires en silicium lors de leur exposition a une dose extrême de radiations. Finalement, un modèle simple de digitalisation à utiliser pour la simulation de performances détecteurs pixels individuels exposés à des faisceau de haute énergie ou bien de l’ensemble du détecteur interne est présenté. Ce modèle simple permets la comparaison entre les données obtenue en faisceau test aux modèle de transport de charge inclut dans ladigitalisation. Le dommage dû à la radiation , l’amincissement et l’utilisation de structures à bords minces sont autant de structures dont les effets sur la collecte de charges affectent les performance du détecteur. Le modèle de digititalisation fut validé pour un détecteur non-irradié en comparant les résultats obtenues avec les données acquises en test faisceau de haut énergie. Le modèle validé sera utilisé pour produire la première simulation de l’IBL incluant les effets d’amincissement du substrat, de dommages dûes aux radiations et de structure dotés de bords fins.In this work, is presented a study, using TCAD simulation, of the possible methods of designing of a planar pixel sensors by reducing their inactive area and improving their radiation hardness for use in the Insertable B-Layer (IBL) project and for SLHC upgrade phase for the ATLAS experiment. Different physical models available have been studied to develop a coherent model of radiation damage in silicon that can be used to predict silicon pixel sensor behavior after exposure to radiation. The Multi-Guard Ring Structure,a protection structure used in pixel sensor design was studied to obtain guidelines for the reduction of inactive edges detrimental to detector operation while keeping a good sensor behavior through its lifetime in the ATLAS detector. A campaign of measurement of the sensor’s process parameters and electrical behavior to validate and calibrate the TCAD simulation models and results are also presented. A model for diode charge collection in highly irradiated environment was developed to explain the high charge collection observed in highly irradiated devices. A simple planar pixel sensor digitization model to be used in test beam and full detector system is detailed. It allows for easy comparison between experimental data and prediction by the various radiation damage models available. The digitizer has been validated using test beam data for unirradiated sensors and can be used to produce the first full scale simulation of the ATLAS detector with the IBL that include sensor effects such as slim edge and thinning of the sensor.","abstract_source_suggest":{"input":"dart-europe.org"}}],"titles":[{"title":"Study of planar pixel sensors hardened to radiations for the upgrade of the ATLAS vertex detector"}],"imprints":[{"date":"2011-06-10"}],"thesis_info":{"institutions":[{"curated_relation":true,"record":{"$ref":"https://inspirehep.net/api/institutions/963424"},"name":"U. Paris-Sud 11, Dept. 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It is obvious that, in such an approximation, radiative corrections—for a given beam energy E0 and a given invariant mass M produced—become independent of the specific process γγ→X considered. The invariant-mass spectrum, corrected for radiation, will be written in the general form dσcorrdM=(1+δ)dσ0dM where dσ0dM is the uncorrected spectrum. Values obtained for δ at typical beam energies E0=1.5, 3, 15, and 70 GeV, and for ME0 ranging between 0.1 and 0.6, are systematically of the order of less than ± 1%.","abstract_source_suggest":{"input":"APS"}}],"refereed":true,"titles":[{"title":"Double equivalent photon approximation including radiative corrections for photon-photon collision experiments without electron tagging"},{"title":"PHOTON PHOTON"}],"external_system_identifiers":[{"schema":"OSTI","value":"6635037"},{"schema":"ADS","value":"1981PhRvD..23..663D"},{"schema":"SPIRES","value":"SPIRES-4972384"}],"facet_author_name":["2324900_M. Defrise","1014509_Christian Carimalo","1019314_Saro Ong","1067027_Jose Manuel Silva"],"_oai":{"sets":["Literature"],"id":"oai:inspirehep.net:163223","updated":"2023-03-06T16:56:22.339425"},"curated":true,"journal_title_variants":["Phys.Rev.D","Phys. Rev. D"]},"updated":"2023-03-06T16:56:22.339425+00:00","created":"2002-06-17T00:00:00+00:00","id":"163223","links":{"bibtex":"https://inspirehep.net/api/literature/163223?format=bibtex","latex-eu":"https://inspirehep.net/api/literature/163223?format=latex-eu","latex-us":"https://inspirehep.net/api/literature/163223?format=latex-us","json":"https://inspirehep.net/api/literature/163223?format=json","json-expanded":"https://inspirehep.net/api/literature/163223?format=json-expanded","cv":"https://inspirehep.net/api/literature/163223?format=cv","citations":"https://inspirehep.net/api/literature/?q=refersto%3Arecid%3A163223"}},{"metadata":{"documents":[{"filename":"10.3929_ethz-b-000402802.pdf","attachment":{"content":"diss . eth no. 26394\n\nS I M U L AT I O N A N D E X P E R I M E N TA L V E R I F I C AT I O N\nO F T R A N S V E R S E A N D L O N G I T U D I N A L\n\nC O M P R E S S I O N O F P O S I T I V E M U O N B E A M S\n\nTowards a novel high-brightness low-energy muon beam-line\n\nA thesis submitted to attain the degree of\ndoctor of sciences of eth zurich\n\n(Dr. sc. ETH Zurich)\n\npresented by\n\nivana belosevic\n\nmag. phys. of University of Zagreb\nborn on 17.05.1990\n\ncitizen of Croatia\n\naccepted on the recommendation of\n\nProf. Dr. K. S. Kirch\nProf. Dr. C. Grab\n\nDr. M. Doser\n\n2019\n\n\n\nIvana Belosevic: Simulation and experimental verification of transverse and lon-\ngitudinal compression of positive muon beams, Towards a novel high-brightness\nlow-energy muon beam-line, © September 2019\n\n\n\nA B S T R A C T\n\nExperiments with positive muons (µ+) and muonium atoms (µ+e−) offer\nseveral promising possibilities for testing fundamental symmetries and the-\noretical predictions of particle physics with high precision. Examples of such\ntests include the search for the muon electric dipole moment, measurement\nof the muon g − 2 and muonium laser spectroscopy. These fundamental\nparticle physics experiments and also solid state investigations using the\nµSR (muon spin rotation) technique could benefit from a high-quality muon\nbeam at low energy with small transverse size and high intensity.\n\nAt the Paul Scherrer Institute, a novel device (muCool) that produces such\na high-quality muon beam is under development, reducing the phase space\nof a standard µ+ beam by 10 orders of magnitude with 10−3 efficiency. The\nphase space compression is achieved by stopping a standard µ+ beam in a\ncryogenic helium gas and subsequently manipulating the stopped µ+ into a\nsmall spot using complex electric and magnetic fields in combination with\ngas density gradients. Such manipulation is done in few successive stages:\ntransverse compression stage, longitudinal compression stage and final com-\npression stage. Finally, muons are extracted through a small orifice into the\nvacuum and into a field-free region. The whole process takes less than 10 µs,\nwhich is essential due to the short muon lifetime of 2.2 µs.\n\nThe compression has been demonstrated for the first two stages individu-\nally. The transverse muon beam compression was demonstrated for the first\ntime in 2015, while the longitudinal compression was demonstrated already\nin 2011, with improved measurements in 2014 and 2015, also including Ê× B̂\ndrift. For construction of these stages it was necessary to define the precise\nand complex electric fields. A method for designing electrodes to produce\nelectric fields of an arbitrary shape has been developed for this purpose.\nTo interpret the measurements, the standard Geant4 packages have been\nextended to include low-energy muon-He interactions, starting from appro-\npriately scaled proton-He cross sections. A good agreement between simu-\nlations and measurements has been observed. The total efficiency of 10−3\n\nthat was predicted from the simulation seems therefore attainable. Recently,\na new concept that combines the first two stages of the original proposal\n(transverse and longitudinal compression stages) into a single mixed com-\npression stage has been developed. In measurements in 2017, the feasibility\nof such a mixed transverse-longitudinal compression stage has been demon-\nstrated.\n\niii\n\n\n\n\n\nZ U S A M M E N FA S S U N G\n\nExperimente mit positiven Myonen (µ+) und Myoniumatomen (µ+e−) bie-\nten diverse vielversprechende Möglichkeiten zur präzisen Überprüfung der\nfundamentalen Symmetrien der Teilchenphysik. Darunter fallen zum Bei-\nspiel die Suche nach dem elektrischen Dipolmoment des Myons, die Mes-\nsung des myonischen g− 2 Faktors und Laserspektroskopie von Myoniuma-\ntomen. All diese Teilchenphysikexperimente zur Grundlagenforschung so-\nwie auch die Methode der Myon-Spin-Rotation (µSR) in der Festkörperphy-\nsik könnten von einem qualitativ hochwertigen Myonenstrahl von niedriger\nEnergie, hoher Intensität und kleinem Strahldurchmesser profitieren.\n\nAm Paul Scherrer Institut wird zur Zeit ein Gerät namens muCool entwi-\nckelt, welches einen solchen Myonenstrahl bereit stellen soll. Simulationen\nzufolge soll der Phasenraum eines herkömmlichen Myonenstrahls bei einem\nWirkungsgrad von 10−3 um zehn Grössenordnungen verkleinert werden.\nDie Kompression des Phasenraums wird erreicht, indem der herkömmliche\nµ+-Strahl in gasförmigem Helium bei kryogenen Temperaturen gestoppt\nwird. Anschliessend werden die Myonen mittels einer komplexen Anord-\nnung von elektrischen und magnetischen Feldern sowie einem Heliumdich-\ntegradienten in einem Punkt gesammelt. Dies geschieht in drei Stufen: Trans-\nversalkompression, Longitudinalkompression und finale Kompression. Ab-\nschliessend werden die Myonen durch eine kleine Öffnung extrahiert und\ngelangen dadurch in eine feldfreie Region im Vakuum. Der gesamte Prozess\ndauert weniger als 10 µs, was wegen der Myonenlebensdauer von nur 2.2 µs\nvon grösster Wichtigkeit ist.\n\nDie Kompression in den ersten zwei Stufen wurde jeweils individuell\nnachgewiesen. Die Transversalkompression wurde erstmals 2015 experimen-\ntell bestätigt, während die Longitudinalkompression bereits 2011 gelang.\nDiese wurde in den Jahren 2014 und 2015 durch Verbesserungen am Gerät\noptimiert, wobei auch die Ê × B̂ Drift berücksichtigt wurde. Für die Kon-\nstruktion der beiden Stufen mussten die komplexen elektrischen Felder prä-\nzise definiert werden. Eigens dafür wurde eine Methode entwickelt, mit der\ndie Elektrodenkonfiguration zu einem beliebigen elektrischen Feld berech-\nnet werden kann. Um die Messungen richtig zu deuten wurden die stan-\ndardmässigen Geant4-Bibliotheken um niederenergetische Myon-Helium In-\nteraktionen erweitert. Ansatzpunkt dafür waren passend skalierte Proton-\nHelium Wirkungsquerschnitte. Die Resultate aus den Simulationen stimmen\nmit den Messungen weitestgehend überein. Der von der Simulation voraus-\ngesagte Wirkungsgrad von 10−3 scheint daher plausibel erreichbar. Kürzli-\nche Entwicklungen eines neuen Konzeptes kombinieren die ersten zwei Stu-\nfen (Transversal- und Longitudinalkompression) der ursprünglichen Idee zu\neiner einzelnen, gemischten Kompressionsphase. In einer Messung wurde\n2017 die Machbarkeit dieser gemischten transvers-longitudinalen Kompres-\nsion nachgewiesen.\n\nv\n\n\n\n\n\nC O N T E N T S\n\n1 introduction 1\n\n1.1 Working principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n\n1.2 Status of the muCool project . . . . . . . . . . . . . . . . . . . . 9\n\n1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 10\n\n2 theory of muon-helium collisions 13\n\n2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13\n\n2.2 Classical scattering theory . . . . . . . . . . . . . . . . . . . . . . 17\n\n2.3 High-energy regime . . . . . . . . . . . . . . . . . . . . . . . . . 29\n\n2.4 Low-energy regime . . . . . . . . . . . . . . . . . . . . . . . . . . 34\n\n2.5 Summary and validity of the classical scattering theory . . . . . 51\n\n3 geant4 simulations of the muon stopping and drift in\n\nhelium gas 57\n\n3.1 Implementation of low-energy muon-He interactions in Geant4 57\n\n3.2 Muon range and slowing-down in He gas . . . . . . . . . . . . . 68\n\n3.3 Muon drift in an uniform electric field . . . . . . . . . . . . . . . 73\n\n3.4 Muon drift in crossed electric and magnetic fields . . . . . . . . 81\n\n3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85\n\n4 transverse compression 89\n\n4.1 Setup for the transverse compression . . . . . . . . . . . . . . . 89\n\n4.2 Results of the 2015 beam-time . . . . . . . . . . . . . . . . . . . . 94\n\n4.3 Electric field scan . . . . . . . . . . . . . . . . . . . . . . . . . . . 105\n\n4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115\n\n5 longitudinal compression 117\n\n5.1 Improved test of the longitudinal compression . . . . . . . . . 117\n\n5.2 Additional muon losses? . . . . . . . . . . . . . . . . . . . . . . . 128\n\n5.3 Longitudinal compression with E×B drift . . . . . . . . . . . . 133\n\n5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138\n\n6 mixed transverse-longitudinal compression 141\n\n6.1 Working principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 141\n\n6.2 Electric field design . . . . . . . . . . . . . . . . . . . . . . . . . . 143\n\n6.3 Measurements and results of the 2017 beam-time . . . . . . . . 152\n\n6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163\n\n7 next steps 165\n\n7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165\n\n7.2 Target with cold extension . . . . . . . . . . . . . . . . . . . . . . 167\n\n7.3 Target with warm extension . . . . . . . . . . . . . . . . . . . . . 176\n\n7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179\n\n8 summary and outlook 187\n\nAppendix 189\n\na motion of charged particles in crossed electric and\n\nmagnetic fields in the vacuum 191\n\nb elastic collisions 193\n\nvii\n\n\n\nviii contents\n\nc elastic scattering in quantum mechanics 197\n\nd electrode design for the mixed transverse-longitudinal\n\ncompression 199\n\nacknowledgments 205\n\nbibliography 207\n\n\n\n1\nI N T R O D U C T I O N\n\nExperiments with positive muons (µ+) enable testing the fundamental sym-\nmetries of particle physics and theoretical predictions with high precision.\nExamples of such tests include the measurement of the anomalous muon\nmagnetic moment (g− 2), searches for the muon electric dipole moment and\nsearches for charged lepton flavor violation in forbidden muon decays. A\ncomprehensive overview of ongoing and future experiments in that direc-\ntion can be found in [1]. In solid state physics, muons are used to probe the\nmagnetic structure of materials via a technique called muon spin rotation\n(µSR) [2].\n\nMoreover, by capturing an electron, a positive muon can form a hydrogen-\nlike bound system called muonium (Mu = µ+e−). A unique aspect of the\nmuonium atom is that it consists only of two leptons and thus the hadronic\n(finite-size) effects are not present. The spectroscopic measurements of the\nMu hyperfine structure and 1s-2s transition serve thus as a precise test of\nbound-state QED1 and fundamental symmetries of the standard model (e.g.,\nthe charge equality between the first and second generation leptons), and\nenable extraction of several fundamental constants, for example, the muon\nto electron mass ratio, the muon magnetic moment and the fine structure\nconstant [3–5]. Besides spectroscopy, experiments looking for muonium-\nantimuonium conversion [6] (a lepton flavor violating process), or measur-\ning the gravitational acceleration of antimatter could also be realized with\nmuonium [7].\n\nTo perform these experiments, muon (and muonium) beams of low en-\nergy and high quality are needed. Standard muon beams are produced at\nproton accelerator facilities, such as PSI2, TRIUMF3 and J-PARC4, among oth-\ners. When accelerated protons hit a production target, they produce pions\nthrough the hadronic interaction, which then decay to muons via the weak\ninteraction. Depending on the position of the pion decay we can distinguish\nroughly between three types of positive muon beams [8]: cloud, surface and\nsub-surface beams. The cloud muons originate from pions decaying in flight,\nwhile surface and sub-surface muons are produced from the pions stopped\nin the production target. Surface muon beams come from the pions decaying\nclose to the surface of the target and are thus monochromatic (29.8 MeV/c\nmomentum) and 100% polarized. Sub-surface muons, on the other hand,\noriginate from pions decaying in the bulk of the target, and thus can have\na range of momenta from about 10 to 29.8 MeV/c. As rates of sub-surface\nmuons decrease rapidly with momentum (a p3.5 dependence is typically ob-\n\n1 Quantum electrodynamics.\n2 Paul Scherrer Institute\n3 Tri-University Meson Facility\n4 Japan Proton Accelerator Research Complex\n\n1\n\n\n\n2 introduction\n\nserved), a high-intensity muon beam with momentum < 10 MeV/c cannot\nbe realized simply by collecting the low-energy muons from the production\ntarget. Thus, special techniques of cooling and moderation are needed.\n\nAt the high-intensity frontier, where the high statistics combined with thin\ntargets are needed, surface muon beams are typically used. Yet, several parti-\ncle physics experiments and the µSR community would significantly benefit\nfrom a slow muon beam with considerably smaller phase space (transverse\nsize, momentum spread and divergence) [9, 10], but still of adequate inten-\nsity. Such a beam would also enable the production of intense muonium\nbeams.\n\nLow-energy µ+ beams can be in principle obtained by moderating a stan-\ndard surface muon beam, but at the cost of beam quality (energy spread,\nsize and divergence). Following the moderation, the beam quality could\nin principle be improved by beam cooling techniques. However, due to the\nshort muon lifetime (τµ = 2.197 µs), standard beam cooling techniques, such\nas stochastic cooling [11] and electron cooling [12], are not suitable for muon\nbeams. Alternative cooling methods producing slow (keV range) muon\nbeams have been developed, employing frictional cooling [13] or muon mod-\neration in frozen rare gas layers [14]. Another technique, based on reso-\nnant laser ionization of muonium, is being developed [15]. However, all of\nthe existing cooling methods can only achieve relatively low beam intensity\n(brightness), due to the low overall cooling efficiencies and due to the fact\nthat they reduce only the energy (momentum) spread of the muon beam,\nwithout compressing the beam transverse size.\n\nThis thesis is devoted to the development of a ultra-slow positive muon\nbeam of high-brightness, as proposed in [16], that addresses some of the\ndeficiencies of the existing muon beam cooling schemes. This novel method\nis based on moderating a standard surface (or sub-surface) muon beam in\nhelium gas, while simultaneously actively reducing the beam transverse size.\nIn this way, the 6D phase space of a muon beam is compressed by 10 orders\nof magnitude with the total compression efficiency of 10−3, thus increasing\nthe muon beam brightness by 7 orders of magnitude.\n\nFigure 1: Sketch of the transformation of a standard muon beam after passing\nthrough the muCool device. For the input beam characteristics, the PSI\nπE5 surface muon beam values have been used.\n\nAfter the compression, the eV energy muon beam is re-accelerated to keV\nenergies, resulting in a pulsed (tagged) beam of 0.1% energy spread, sub-mm\nsize and divergence of about 30 mrad (see Fig. 1). If, for example, the πE5\n\n\n\n1.1 working principle 3\n\nsecondary muon beam at PSI, providing about 2 · 108 µ+/s (at 29.8 MeV/c)5,\nwould be used as an input beam, the muCool device could deliver about\n2 · 105 µ+/s with close to 100% polarization.\n\n1.1 working principle\n\nThe device [16] being developed by the muCool collaboration, consists of\nseveral stages, as sketched in Fig. 2. The whole device is placed inside an\n5 T magnetic field, pointing in the +z-direction. First, a surface muon beam\nof 29.8 MeV/c momentum propagating in the −z-direction is stopped in few\nmbar of helium gas at cryogenic temperature, reducing the muon energy\nfrom 4 MeV to the several eV range. The “stopped” muons are then ma-\nnipulated using a combination of complex electric and magnetic fields and\ndensity gradients. In the first stage, the muon beam size is reduced in the y-\ndirection (transverse compression). After that, the muons are transported to\nthe second stage, which is at room temperature, where the muon beam size\nis reduced also in the z-direction. Subsequently, the muons are extracted\nfrom the He gas into vacuum, re-accelerated to keV energy with a pulsed\nelectric field and extracted out of the magnetic field.\n\nThe extracted beam has sub-mm transverse size and an eV energy spread\nin both the longitudinal and the transverse directions. As a result, the phase\nspace of a surface muon beam is reduced by about 10 orders of magnitude.\nThe whole process takes about 10 µs, thus yielding a total compression effi-\nciency of 10−3, mainly limited by the short muon lifetime (τµ = 2.197 µs).\n\nIn the following, the working principle of each stage will be explained in\nmore detail.\n\ntransverse compression stage In the transverse compression stage,\nthe secondary (surface) µ+ beam is stopped in a few mbar of helium gas\nat cryogenic temperatures. The muon stopping distribution has a radius\nof about 1 cm in the xy-plane and is elongated along the beam (z-) axis6.\nThe size of the stopping volume is determined by the initial energy and\ntransverse size of the secondary muon beam and the gas density. In this\nstage, the magnetic field of 5 T points in the +z-direction and the electric\nfield is at 45° with respect to the x-axis:\n\nE =\n\nExEy\n0\n\n , (1)\n\nwith Ex = Ex ≈ 1 kV/ cm.\nTo understand the working principle of the transverse compression stage,\n\nwe need to first understand the motion of the muon in the helium gas when\n\n5 This number includes a projected increase of the muon rates by 30%, arising from the opti-\nmization of the production target geometry (slanted target), to be tested in November 2019.\n\n6 The length in the z-direction will depend on possible degraders placed upstream of the mu-\nCool device to pre-moderate the beam. The typical length of the muCool device is expected\nto be about 1 m.\n\n\n\n4 introduction\n\nFigure 2: Scheme of the muCool device, reproduced from [17]. A standard surface\nmuon beam is stopped in cryogenic helium gas with a vertical temperature\ngradient inside a 5 T magnetic field. The size of the stopped muon swarm\nis then compressed using complex electric fields and density gradients in\nseveral stages: first in the y-direction (transverse compression), then in the\nz-direction (longitudinal compression). After those two stages, the muon\nswarm size is compressed further in the final compression stage, featuring\na combination of transverse and longitudinal compression. Finally, the\nmuons of a sub-mm swarm diameter are extracted through a small orifice\ninto vacuum, where they are re-accelerated along the z-axis, extracted\nfrom the 5 T field and sent to another experiment.\n\ncrossed electric and magnetic fields are applied. We start by considering the\nmuon motion under the influence of such fields in the vacuum, which will\nhelp to later analyze the muon motion in the gas.\n\nIn vacuum, applying such crossed electric and magnetic fields would\nprompt the stopped muons to drift in the Ê × B̂-direction, as sketched in\nFig. 3. Such drift can be understood in the following way: initially (at po-\nsition 1 in Fig. 3), the muon is at rest and is thus not influenced by the\nmagnetic field. Due to the electric field, the muon accelerates in the electric\nfield direction. As it gains velocity, its trajectory will be bent by the magnetic\nfield. At a certain point, after the position 2 has been passed, the bending of\nthe muon trajectory due to the magnetic field will cause the muon to move\nagainst the electric field, so that the muon is decelerated, and eventually\ncoming at rest again, at position 3. At this point, the total muon displace-\nment in the Ê-direction is 0, but the muon has traveled some distance in the\nÊ× B̂-direction. This kind of cycloidal motion is repeated periodically, with\nfrequency ω = eB/mµ (cyclotron frequency), where mµ is the muon mass.\n\nThe trajectory of the muon in vacuum can be obtained by integrating the\nequations of motion (see Appendix A). To simplify the description of the\nmuon motion, we introduce the x ′−y ′ coordinate system, where the y ′-axis\nis parallel to the Ê-direction and the x ′-axis is parallel to the Ê× B̂-direction.\nUsing this reference system, the y ′- and x ′-positions of the muon versus\ntime in vacuum are:\n\n\n\n1.1 working principle 5\n\nFigure 3: The cycloidal motion of a positive muon in vacuum with crossed electric\nand magnetic fields applied. Initially, the muon is at rest at position 1, but\nunder the influence of the electric and magnetic fields, it drifts on average\nin the Ê× B̂-direction.\n\ny ′ (t) =\nE\n\nB\n\n1\n\nω\n(1− cos(ωt)) (2)\n\nx ′ (t) =\nE\n\nB\n\n1\n\nω\n(ωt− sin(ωt)) . (3)\n\nHowever, our muCool device is filled with helium gas, resulting in muon\ncollisions with the He gas atoms at an average frequency νc = 1/τ (τ is\nthe mean free time between two collisions), that depends on the gas den-\nsity, muon-He elastic collision cross sections and muon-He relative velocity.\nThese collisions modify the muon motion compared to the vacuum case. If\nwe assume that the muon loses all of its energy at each collision, between\ntwo consecutive collisions the muon moves according to the Eqs. (2) and (3) 7\n\n(with the origin of the coordinate system defined by the position of the last\ncollision).\n\nIf the collision frequency νc is higher than the cyclotron frequency ω, the\nmuon cycloidal motion will be “reset” before the magnetic field bends the\nmuon trajectory down to the y ′ = 0, as sketched in Fig. 4. The net result\nis that the muon drift velocity acquires a component in the Ê-direction, and\nnot just the Ê× B̂-direction, as in vacuum. The deflection angle θ from the\nvacuum value (from the Ê× B̂-direction) is proportional to the collision fre-\nquency: for high collision frequencies (compared to the cyclotron frequency)\nthe deflection angle θ is large (blue trajectory), while for low collision fre-\nquencies θ is small (green trajectory).\n\nThe deflection angle θ can be defined as the ratio of the average velocities\nin the y ′- and in the x ′-direction:\n\ntan θ =\nvy ′\n\nvx ′\n, (4)\n\nwhere the average velocities are calculated from the average distances trav-\neled in the y ′- and the x ′-direction between two collisions, divided by the\nmean free time between two collisions τ:\n\nvy ′ = y\n′/τ (5)\n\nvx ′ = x\n′/τ. (6)\n\n7 The same will be true on average if the scattering is isotropic. Indeed the random component\nof the velocity after the collision averages to zero over many collisions.\n\n\n\n6 introduction\n\nFigure 4: Schematic of two muon trajectories in a gas with crossed electric and mag-\nnetic fields applied. Initially, the muons are at rest at the origin of the\ncoordinate system, but under the influence of electric and magnetic fields,\nthey start moving as in vacuum (see Fig. 3). However, their cycloidal mo-\ntion will be interrupted by collisions with He gas atoms (represented by\nthe yellow stars). If the muon is stopped in the collision, its cycloidal mo-\ntion will be “reset”. These collisions deflect the muon drift direction from\nthe Ê× B̂-direction by an angle θ. The deflection is large for large collision\nfrequency (blue trajectory) and small for low collision frequency (green\ntrajectories).\n\nThe average distances y ′ and x ′ can be obtained from Eq. (2) and (3) by\naveraging over many collisions:\n\ny ′ =\n\n∞∫\n0\n\ny ′(t)f(t)dt (7)\n\nx ′ =\n\n∞∫\n0\n\nx ′(t)f(t)dt, (8)\n\nwhere f(t) is the probability that the free time between two collisions is\nbetween t and t+ dt, which is described by the exponential distribution:\n\nf(t) =\n1\n\nτ\ne−t/τ. (9)\n\nThus, Eqs. (7) and (8) become:\n\ny ′ =\nE\n\nB\n\n1\n\nωτ\n\n∞∫\n0\n\n(1− cos(ωt)) e−t/τdt =\nE\n\nB\n\nωτ\n\n1+ω2τ2\n(10)\n\nx ′ =\nE\n\nB\n\n1\n\nωτ\n\n∞∫\n0\n\n(ωt− sin(ωt)) e−t/τdt =\nE\n\nB\n\nω2τ2\n\n1+ω2τ2\n, (11)\n\nand the deflection angle of Eq. (4) is:\n\ntan θ =\ny ′\n\nx ′\n=\n\n1\n\nωτ\n=\nνc\n\nω\n. (12)\n\n\n\n1.1 working principle 7\n\nWe can see that the deflection angle increases with the collision frequency\nνc, as predicted earlier. Therefore, we can manipulate the direction of the\nmuon drift by changing the collision frequency, which is given by:\n\nνc = NσEL(vr)vr, (13)\n\nwhere N is the helium gas density, σEL is the cross section for elastic colli-\nsions of muons with helium atoms and vr is the relative velocity between\nmuons and He atoms. The most straightforward way to modify the collision\nfrequency is by changing the gas density N. The manipulation of the muon\ndrift direction is at the core of the muCool compression scheme, allowing us\nto steer muons which have been stopped at different locations in the target\ntowards a single location.\n\nFigure 5: Sketch of the transverse compression stage. The bottom wall of this stage\nis kept at 4 K, while the top wall is at 12 K, creating a temperature gradi-\nent in the y-direction. The electric field is at 45° to the x-axis, while the\nmagnetic field points in the +z-direction. The y ′ − x ′ coordinate system,\nused in the text is also sketched. The y ′-axis of this system is aligned with\nthe electric field directions, while x ′-axis points in the Ê× B̂-direction. The\ntrajectories of three muons, stopped in regions of different gas density of\nthe transverse stage are also sketched: they move from left to right (in the\n+x-direction), while converging in the y-direction.\n\nIn the transverse compression stage, this is done by creating a temperature\ngradient in the vertical (y-) direction. As sketched in Fig. 5, the bottom of this\nstage is kept at 4 K, while the top is at 12 K, resulting in a vertical density\ngradient, as shown by the color gradient in Fig. 5. Muons at different y-\npositions (at fixed x-position) experience thus different densities, resulting\nin different drift directions. This effect is shown for three muon trajectories,\nstarting at three different y-positions, indicated by the full colored circles\nin Fig. 5. The coordinate system aligned with the electric field direction\n(y ′ − x ′) is also sketched in this figure. This coordinate system is rotated by\n−45° around the z-axis of the y− x coordinate system. In this case, the\n\ndensity is adjusted\nby changing the\nhelium gas\npressure, while\nkeeping a fixed\ntemperature\ngradient from 4 to\n12 K.\n\nWe start by considering a muon stopped at y = 0 (mid-height of the target,\nmarked by the full gray circle in Fig. 5). The gas density is chosen so that\nνc/ω = 1 at this position. According to Eq. (12), the deflection angle with\n\n\n\n8 introduction\n\nrespect to x ′-axis is thus 45°, i.e., the muon drifts in the +x-direction (see\ngray trajectory in Fig. 5).\n\nNext, we examine the drift of a muon stopped in the bottom part of the\ntarget (full blue circle in Fig. 5). In this part of the target, the temperature\nis lower (the gas density is higher) relative to the mid-plane of the target.\nTherefore, the collision frequency is higher than the cyclotron frequency:\nνc/ω > 1. This means that the deflection angle θ from the x ′-axis is large,\nso that muon drifts predominantly in the y ′-direction (upwards), as sketched\nby the blue trajectory in Fig. 5.\n\nFinally, we consider the drift of a muon stopped in the upper part of the\ntarget (full green circle in Fig. 5). In this case, the gas density is lower than\nat y = 0, so that νc/ω < 1. The deflection angle θ is thus small, resulting in\nmuon drift predominantly in the x ′-direction (downwards).\n\nThe net result (as demonstrated on the example the 3 trajectories con-\nsidered above) is that the muons stopped at different y-positions converge\ntowards y = 0, while simultaneously drifting in the +x-direction. Hence, the\nsize of the muon swarm8 is reduced in the y-direction (transverse compres-\nsion). This compression stage takes between 1 and 3 µs: the muons traveling\nthrough higher density are slower than the muons moving through the layer\nof lower gas density.\n\nlongitudinal compression stage : In the transverse compression\nstage, the muon swarm size is reduced in the y-direction, but the muon\nswarm is still elongated in the z-direction. To reduce the muon swarm size\nin the z-direction, muons are guided from the transverse compression stage\nto the longitudinal compression stage, which is at room temperature. In this\nstage, an electric field of the form\n\nE =\n\n\n0\n\nEy\n\n−Ez · z|z|\n\n , (14)\n\nis applied inside the gas, with typically Ey = 2 ·Ez ≈ 0.2 kV/ cm. In contrast\nto the electric field in the transverse compression stage (see Eq. (1)), the elec-\ntric field here also has a component parallel to the magnetic field, pointing\ntowards the center of the target at z = 0, in addition to the component (Ey)\nperpendicular to the magnetic field (see Fig. 2).\n\nDue to the electric field component parallel to the magnetic field, the\nmuon drift velocity has a component pointing in the longitudinal direction,\ntowards the center of the target:\n\nvz = µ̃Ez (15)\n\n8 We introduce here the concept of a muon swarm: even though we typically use the muCool\ndevice with continuous secondary muon beams, so that we only have one muon at the\ntime inside the device, the drift of many such individual muons can be thought of as a\nsimultaneous movement of a muon ensemble (swarm). These two descriptions of the muon\ndrift are statistically equivalent.\n\n\n\n1.2 status of the mucool project 9\n\nwhere µ̃ = e/νcmµ is muon mobility. Therefore, the muon spread in the\nz-direction is reduced (longitudinal compression). Under certain conditions\n(large enough Ez/N) the muon mobility can suddenly increase (so-called\nrunaway effect [18]), resulting in a large vz velocity and thus a rapid con-\nvergence of the muons towards the z = 0. This effect allows efficient longi-\ntudinal compression of relatively long muon swarm (100 mm), within only\n2 µs.\n\nThe perpendicular component of the electric field (Ey) at this low density\nsimply causes the drift velocity to have a component in the Ê× B̂-direction,\nwhich corresponds here to the +x-direction. In this way, the muons are\ntransported towards the final compression stage and the vacuum extraction\nstage (see Fig. 2), while simultaneously being compressed in the z-direction.\n\nthe transition stages and the extraction into vacuum : Af-\nter the transverse and longitudinal compression stages, a final compression\nat cryogenic temperature is occurring, both in y- and z- directions. In this\nway the beam size is further reduced, so that the muons can be efficiently\nextracted into vacuum through an orifice of 1 mm diameter. To keep the\nthe gas density distribution inside the target stable, without flow of gas or\nturbulences that would disrupt the required density gradients, the helium\ngas needs to be continuously injected into the target at the orifice location9.\nThe muons leave the gas target through the orifice and enter into the vac-\nuum. After the extraction, they are re-accelerated along the z-direction to\nkeV energies and transported to a field-free region.\n\nNote that, similar to the final compression stage, also between the trans-\nverse and the longitudinal compression stages there is a transition stage,\nfeaturing complex electric fields, which are a combination of the fields re-\nquired for a simultaneous compression in the transverse and longitudinal\ndirection.\n\n1.2 status of the mucool project\n\nInitially, various stages of the muCool compression scheme can be tested\nindependently. The first demonstration of the longitudinal compression was\nachieved in the beam-time 2011 [19]. However, several problems prevented\na quantitative extraction of the compression efficiency. These problems were\nsolved in the beam-time in 2014, where an improved test of the longitudinal\ncompression was performed. In the next beam-time, in 2015, the full lon-\ngitudinal compression stage was tested, including also the Ê× B̂ drift. The\nresults of these two measurements have been published in [20].\n\nIn parallel, the development of the transverse compression stage was on-\ngoing. In 2015, we succeeded in developing the challenging cryogenic target\nrequired for this stage (for more information on the development see [21,\n22]). During the beam-time of 2015, the transverse compression of the muon\nbeam was demonstrated for the first time.\n\n9 A scheme has been devised by the muCool collaboration, but it is not presented in this thesis.\n\n\n\n10 introduction\n\nFollowing the successful separate tests of the first two beam-line stages, we\nhave explored the possibility of combining the longitudinal and transverse\ncompression stage into a single mixed transverse-longitudinal compression\nstage, as a simpler, and possibly more efficient, alternative to the scheme pre-\nsented in this chapter, based on [16]. This mixed compression stage could be\nalso used as a simpler setup to test the muon extraction through the small\norifice. The development of the mixed compression stage was undertaken\nin 2017. In the preliminary measurements of the beam-time 2017, a mixed\ntransverse-longitudinal compression of the muon beam was observed for the\nfirst time, but limited by the maximal electric field we were able to achieve.\nPresently, the muCool collaboration is making an effort to improve the elec-\ntric field strength, and thus also the efficiency of compression. The improved\ntest of the mixed compression stage is planned at the end of 2019.\n\nIn order to correctly design and optimize the various compression stages,\na Geant4 simulation of the muon drift through our device is needed. For that\npurpose, low-energy muon physics processes were implemented in Geant4,\nmainly low-energy elastic collisions and charge exchange. Since typically the\ncross sections needed to describe the muon-He interactions are not known,\nthey have been scaled from proton data. The simulation based on these cross\nsections has been verified by the good agreement between its predictions and\nvarious compression stage measurements. Therefore, all our experimental\nresults, to date, confirm that the proposed (simulated) efficiency of 10−3 is\nattainable.\n\n1.3 outline of the thesis\n\nThe thesis is structured in the following way:\n\n• Chapter 2 outlines the theoretical background required to implement\nthe low-energy muon interactions into the Geant4 simulation. It is\nshown, using classical scattering theory, that it is possible to deduce\nthe needed muon cross sections from the existing proton cross sections\nby appropriate scaling. The limitations of this approach are also dis-\ncussed.\n\n• Chapter 3 applies the conclusions of the Chapter 2 to the actual Geant4\nsimulation. The implementation of the low-energy muon interactions\ninto the Geant4 simulation is described in the beginning of the chap-\nter. This simulation is then used to study the muon slowing-down and\ndrift in three simple cases that are at the foundation of the muCool\ncompression scheme: muons stopping in helium, muons drifting un-\nder the influence of the uniform electric field, and muons drifting in\ncrossed electric and magnetic fields.\n\n• Chapter 4 presents the setup and the measurement results of the trans-\nverse compression test in beam-time 2015. The measurements are inter-\npreted using the Geant4 simulation introduced in Chapter 3 to quantify\nthe efficiency of the compression.\n\n\n\n1.3 outline of the thesis 11\n\n• Chapter 5 reports on the improved measurements of longitudinal com-\npression, with and without Ê × B̂-drift. The comparison with the\nGeant4 simulations is used to extract the compression efficiency of this\nstage. The text of this chapter is largely reproduced from the author’s\npublication [20].\n\n• Chapter 6 describes a concept where the transverse and longitudinal\ncompression stage are combined into a single mixed compression stage.\nThe design optimization of the mixed compression stage was done us-\ning Geant4 simulation, in conjunction with Comsol simulations of the\nhelium gas density and the electric field. The measurements obtained\nin the beam-time of 2017 are also presented and discussed.\n\n• Chapter 7 considers several possible extensions of the mixed transverse-\nlongitudinal compression stage that could allow efficient muon beam\nextraction into the vacuum. The Geant4 simulation based on the realis-\ntic electric field design and density gradients is used to study possible\nimprovements in the compression efficiency.\n\n• To conclude, the summary of the results of this thesis and the possible\ndirections of future developments are discussed in Chapter 8.\n\n\n\n\n\n2\nT H E O RY O F M U O N - H E L I U M C O L L I S I O N S\n\nWhen the surface muon beam enters the muCool device, it is decelerated\nfrom an energy of 4 MeV down to below 100 eV in about 100 ns [23]. After\nthat, the muons are steered through the muCool device for about 10 µs.\nDuring this time, their energies typically range from 1 to 100 eV.\n\nThis slowing-down and drifting processes in the muCool target are gov-\nerned by the muon collisions with the He atoms. In order to model the\nmuon motion in the gas correctly, we need thus to implement the muon-He\ninteractions in the whole energy range from 4 MeV all the way down to\nbelow 1 eV in the Monte-Carlo (Geant4) simulation. Which type of interac-\ntion (ionization, excitation, charge exchange, elastic collisions) is dominant\ndepends on the muon energy.\n\nEach interaction can be described with differential cross sections. Unfor-\ntunately, such data for the muons is mostly not available, especially at low\nenergies. However, to a good approximation, it is possible to use appropri-\nately scaled proton cross sections instead, which are more readily available.\n\nIn this chapter the necessary theoretical background will be summarized,\nwhich will allow us to deduce the scaling between muon-He and proton-\nHe cross sections. Then, the implementation of the most relevant muon-He\ninteractions in the Monte-Carlo simulation will be outlined.\n\n2.1 introduction\n\nWhen a charged particle (muon) passes through a medium (He gas) it inter-\nacts with the target particles via Coulomb interaction. The interaction can\nbe either with the atom as whole or with the electrons or the nucleus of the\natom. These interactions can lead to a change of the projectile velocity, i. e.,\ndeflection from the initial direction and decrease of the speed. Additionally,\nthe collision may also lead to a change in the internal state of the projectile\nand/or target particle.\n\nDepending on the outcome of the interaction, the collision can be classified\neither as elastic or inelastic. The elastic collisions are considered to be those\nthat only lead to a change in projectile direction and kinetic energy, while\nthe internal state of both projectile (A+) and target particle (B) is unaffected:\n\nA+ +B→ A+ +B. (elastic) (16)\n\nInelastic collisions are the collisions that change also the internal state of\nthe projectile and/or target particle. Possible inelastic collisions include tar-\n\n13\n\n\n\n14 theory of muon-helium collisions\n\nFigure 6: Sketch of the four main types of collisions that a positively charged parti-\ncle A+ (muon) experiences when passing through a gas (made of neutral\natoms B): elastic collisions, target atom excitation, target atom ionization\nand electron capture. In a collision, the velocity directions and the mag-\nnitudes, as well as the internal states of the projectile and target can be\nchanged.\n\nget atom excitation and ionization, and capture of a target electron by the\nprojectile:\n\nA+ +B→ A+ +B∗ (excitation) (17)\n\nA+ +B→ A+ +B+ + e− (ionization) (18)\n\nA+ +B→ A+B+ (electron capture) (19)\n\nwhere A = A+e−. These different kinds of collisions are sketched in Fig. 6.\nOnce the projectile is neutralized via reaction (19), it can be re-ionized in\n\nanother collision:\n\nA+B→ A+ +B+ e−. (electron loss) (20)\n\nHowever, before it is re-ionized it can collide elastically or in-elastically with\nthe target atoms. These collisions lead to additional energy losses, which\nalso need to be taken into account.\n\nThe rate of occurrence of these various collision types depends strongly on\nthe incident particle (projectile) energy. Depending on the projectile velocity,\nwe can distinguish roughly between three regimes:\n\nhigh-energy regime , where the projectile velocity is much higher than\nthe orbiting (Bohr) velocity of the most loosely bound electron of the\ntarget atom. In this regime inelastic collisions, such as ionization\nand excitation of the target atoms dominate the energy loss (stopping\npower) of the projectile.\n\n\n\n2.1 introduction 15\n\ncharge-exchange regime , where the projectile velocity is lower than\nthe Bohr velocity of the most loosely bound electron of the target atom.\nIn this regime, the projectile goes through many cycles of electron cap-\nture and subsequent electron loss. Therefore, the projectile (A+) will\nspend significant amount of time while slowing down as neutral par-\nticle (A). For muons, this cyclical charge exchange is the dominant\nmechanism of energy loss from about 2 keV down to about 100 eV.\n\nlow-energy regime , where the elastic collisions with the target atoms\ndominate the interactions between the projectile and the target atoms.\nFor muons, this is the dominant process from about 100 eV down to\nthermal energies.\n\nTo model the muon drift in the muCool device, we need to know the proba-\nbility and the outcome of the collisions (energy loss and deflection) for each\ntype of collision. This information is contained in the double differential\ncross section d2σ\n\nd(∆E)dΩ , which is a function of the projectile energy E, the\ndeflection solid angle Ω and the energy loss ∆E.\n\nThe total probability of a certain interaction occurring is proportional to\nthe integral of the corresponding double differential cross section over all\npossible energy losses and deflection angles:\n\nσ(E) =\n\n∫∫\nd2σ\n\nd(∆E)dΩ\n(Ω,∆E,E)d(∆E)dΩ, (21)\n\nreferred to as the total cross section. Often, the energy loss of the particle is\none-to-one (injective) function of the deflection angle, so that the collision is\ncompletely described by an angular differential cross section dσ\n\ndΩ(Ω,E) only,\nor alternatively by an energy-loss cross section dσ\n\nd(∆E)(∆E,E). Hence, in these\ncases the total cross section simplifies to\n\nσ(E) =\n\n∫\ndσ\n\ndΩ\n(Ω,E)dΩ =\n\n∫\ndσ\n\nd(∆E)\n(∆E,E)d(∆E) =\n\n∫\ndσ(E), (22)\n\nwhere we introduced the simplified notation\n\ndσ(E) ≡ dσ\n\ndΩ\n(Ω,E)dΩ =\n\ndσ\n\nd(∆E)\n(∆E,E)d(∆E). (23)\n\nA further simplification is possible by considering the slowing-down process\naveraged over many collisions. The average energy loss per collision for a\nprojectile of energy E can then be written as:\n\n∆E(E) =\n\n∫\n∆E(E)dσ(E)∫\ndσ(E)\n\n=\n\n∫\n∆E(E)dσ(E)\n\nσ(E)\n. (24)\n\nThe integral in the numerator of Eq. (24) is the so-called stopping cross sec-\ntion or stopping power [24]:\n\nS(E) =\n\n∫\n∆E(E)dσ(E) (25)\n\n= −\n1\n\nN\n\ndE\n\ndx\n(E), (26)\n\n\n\n16 theory of muon-helium collisions\n\nwhich is essentially the average energy loss per unit path length and per\ntarget atom (N is the number density of the gas). Using Eqs. (24) and (25),\nthe stopping power can be expressed as:\n\nS(E) = ∆E(E) · σ(E). (27)\n\nTherefore, the stopping power can be determined from the average energy\nloss ∆E of the projectile at a certain energy E and the total cross section at\nthis energy E.\n\nThe energy loss of the projectile can be computed from the momentum\nand energy conservation laws, while the total cross sections can be taken\neither from measurements or from theoretical calculations. However, such\ncross-section data and theoretical predictions are basically non-existent for\nmuons in helium, especially at low energies. Since the interaction between\nmuons and He atoms is a Coulomb interaction, the interaction potential\nis basically the same as the interaction potential between protons and he-\nlium, because the charge of muon and proton are the same. This similarity\nwill allow us to use the cross sections for protons in He, which are more\nreadily available. However, the muon and proton masses are very different\nmp ' 8.88mµ, so the proton cross sections need to be scaled appropriately\nto account for the mass difference. This scaling also depends on the type\nof the interaction considered - it is indeed different for elastic and inelastic\ncollisions.\n\nTo determine the correct scaling, we need to examine in more detail the\ntheoretical approach used to calculate the cross sections. The general behav-\nior of the cross sections can be inferred already from the classical scattering\ntheory in a relatively straightforward manner. This is the approach we will\nfollow in this chapter. Clearly, the classical approach will fail at some point,\nmainly for very small deflection angles, where a full quantum-mechanical\napproach would be needed. This implies that the scaling relationships be-\ntween muon and proton cross sections derived from the classical scattering\ntheory are not valid anymore for very small angles. Moreover, no such\nscaling is even possible in this regime. This in turn limits the proton cross\nsections we can adopt for muons.\n\nThe outline of the rest of this chapter is the following:\n\n• In Section 2.2, we will first consider the kinematics of the elastic and\ninelastic collisions. This will allows us to calculate the energy loss of\nthe muon in a collision. The rest of this section will summarize the\nclassical method for calculating the differential and total cross sections\nwhen the interaction potential between two particles is known.\n\n• In Section 2.3, the classical scattering theory will be applied to the high-\nenergy regime (> 10 keV) of proton-He collisions. We will then relate\nthese results to muons, deducing the appropriate scaling.\n\n• In Section 2.4, we will study the scaling between proton-He and muon-\nHe cross sections for the low-energy regime (< 10 keV). We will treat\nboth electron-capture and low-energy elastic collisions, first for pro-\ntons, then for muons.\n\n\n\n2.2 classical scattering theory 17\n\n• In Section 2.5, the results for both high-energy and low-energy regime\nwill be summarized, in particular the obtained scaling between proton\nand muon cross sections and stopping power. The limits of validity\nof the classical approach for each type of interaction will also be dis-\ncussed.\n\n2.2 classical scattering theory\n\nIn this section we will first consider the kinematic aspect of the collisions in\norder to get expressions for the energy loss and the scattering angle of the\nprojectile. These are obtained by applying conservation of energy and con-\nservation of momentum to the collision process, first in the laboratory frame,\nthen in the center-of mass frame. The classical scattering theory is then used\nto connect the obtained energy loss and scattering angle of the projectile to\nthe differential cross sections and the stopping power for different collision\ntypes.\n\nIn the following, non-primed variables refer to the variables before col-\nlision, while primed variables refer to the variables after the collision has\ntaken place. The index A refers to the projectile, while B refers to target\natom. In all the equations that follow, we have assumed that the particle B is\nat rest in the laboratory frame before the collision, given the small velocity\nof the He atoms compared to the muons1.\n\nelastic collisions kinematics\n\nCollisions in the laboratory frame A sketch of the collision in the labo-\nratory frame is shown in Fig. 7 (top). Energy conservation in the laboratory\nframe (assuming vB = 0) reads:\n\n1\n\n2\nmAv\n\n2\nA =\n\n1\n\n2\nmAv\n\n′2\nA +\n\n1\n\n2\nmBv\n\n′2\nB . (28)\n\nTherefore, the energy change (loss) of the projectile ∆E in the collision is\nequal to the recoil energy of the target particle, so that the total kinetic energy\nis conserved:\n\n∆E =\n1\n\n2\nmA(v\n\n2\nA − v ′2A ) (29)\n\n=\n1\n\n2\nmBv\n\n′2\nB ≡ T . (30)\n\nFrom now on, we will refer to T as the elastic energy loss.\nSimilarly, the momentum conservation (assuming vB = 0) leads to:\n\nmAvA = mAv ′A +mBv ′B, (31)\n\nwhich can be separated into two components (in the plane of the collision):\n\nmAvA = mAv\n′\nA cos ϑA +mBv\n\n′\nB cos ϑB (32)\n\n0 = mAv\n′\nA sin ϑA −mBv\n\n′\nB sin ϑB. (33)\n\n1 The same assumption is also used in the Geant4 simulation later.\n\n\n\n18 theory of muon-helium collisions\n\nFigure 7: Sketch of a two-particle (A and B) collision in the laboratory frame (top)\nand the center-of-mass frame (middle). The bottom panel shows an equiv-\nalent representation of the collision as a scattering of a single particle in\na central potential, which reduces the two-body problem to a one-body\nproblem.\n\n\n\n2.2 classical scattering theory 19\n\nIn summary, we get three equations (Eqs. (28), (32) and (33)) that define\nthe collision kinematics. We can assume that we know mA, mB and vA,\nso this leaves us with four variables to be determined: v ′A, v ′B, ϑA and ϑB.\nObviously, we cannot determine these four variables from the conservation\nlaws only. We also need to consider the details of the interaction, as we\nwill see later. However, already at this stage, we can relate the two final\nscattering angles (ϑA, ϑB) by combining the previous equations [25]:\n\ntan ϑA =\nsin 2ϑB\n\nmA/mB − cos 2ϑB\n. (34)\n\nHence, for mA � mB (e.g., a muon or a proton scattering off an electron)\nϑA → 0, i.e., the projectile is not strongly deflected.\n\nCollisions in the center-of-mass frame At this point, it is helpful to in-\ntroduce the center-of-mass system, which allows us to simplify the analysis\nof the collisions. The center-of-mass frame moves with the center-of-mass\nvelocity vC with respect to the laboratory frame:\n\nvC =\nmAvA +mBvB\nmA +mB\n\n=\npC\n\nmA +mB\n. (35)\n\nThe velocities of the colliding particles in the center-of-mass frame are pro-\nportional to the relative velocity of the particles vr = vA − vB:\n\nvA,CM = vA − vC =\nmB\n\nmA +mB\nvr =\n\nmr\n\nmA\nvr (36)\n\nvB,CM = vB − vC = −\nmA\n\nmA +mB\nvr = −\n\nmr\n\nmB\nvr, (37)\n\nwheremr = mAmB\n\nmA+mB\nis the reduced mass of the two colliding particles. From\n\nthese equations it follows that the momenta of the two colliding particles in\nthe center-of-mass frame have same magnitude, but opposite direction (so\nthat the total momentum in the center-of-mass frame is 0):\n\npA,CM = −pB,CM = mrvr ≡ pCM. (38)\n\nThis is true at all times during the collision, so it holds for the initial and\nfinal states as well. The total energy in the center-of-mass frame is equal to\nthe relative kinetic energy:\n\nEA,CM + EB,CM =\n1\n\n2\nmrv2r ≡ ECM. (39)\n\nWhen the target particle is at rest, the center-of-mass energy can be simply\nrelated to the projectile energy:\n\nECM =\nmB\n\nmA +mB\nEA. (40)\n\nWe will now examine the collision in the center-of-mass frame, as sketched\nin Fig. 7 (middle). First, we note that, when there are no external forces,\n\n\n\n20 theory of muon-helium collisions\n\nthe center-of-mass velocity does not change in the collision, because of the\nmomentum conservation (Eq. (31)):\n\nv ′C =\nmAv ′A +mBv ′B\nmA +mB\n\n=\nmAvA +mBvB\nmA +mB\n\n= vC. (41)\n\nFrom the conservation of energy it follows that in an elastic collision the rel-\native velocity changes only the direction, while its magnitude is conserved:\n\nECM = E ′CM ⇒ 1\n\n2\nmrv2r =\n\n1\n\n2\nmrv ′2r . (42)\n\nTherefore, in the center-of-mass frame, both particles are deflected by the\nsame angle χ (see Fig. 7 (middle)), which corresponds to the angle between\nthe relative velocity before (vr) and after (v ′r) the collision. It can be shown\nthat the scattering angles in the laboratory (ϑA) and the center-of-mass frame\n(χ) are related by the following expression [25]:\n\ntan ϑA =\nsinχ\n\ncosχ+ mA\n\nmB\n\n. (43)\n\nNote the two limiting cases: when mA � mB, ϑA is small (as shown before).\nWhen mA � mB, the laboratory and the center-of-mass scattering angles\nare approximately the same: ϑA ≈ χ .\n\nThe introduction of the center-of-mass frame allows us to express the elas-\ntic energy loss T (in the laboratory frame) of the projectile as (see Appendix B\nfor the derivation):\n\nT = 2\nmAmB\n\n(mA +mB)\n2\nEA (1− cosχ) (44)\n\n= 2\nmA\n\nmA +mB\nECM (1− cosχ) , (45)\n\nwhere EA = 1\n2mAv\n\n2\nA is the initial projectile energy in the laboratory system\n\nand χ is the scattering angle in the center-of-mass frame. We can see that the\nenergy loss of the projectile is uniquely determined by the scattering angle\nχ in the center-of-mass frame. The maximum possible elastic energy loss is\nobtained for χ = π (backward scattering):\n\nTmax = 4\nmAmB\n\n(mA +mB)\n2\nEA. (46)\n\n\n\n2.2 classical scattering theory 21\n\nIn the inelastic\ncollision leading to\ncharge exchange\n(Eqs. (19) and\n(20)), the masses of\nthe projectile and of\nthe target particle\nare also changed.\nHowever, the\ncharge exchange\nfor muons\n(protons) in helium\nmostly involves\nonly transfer of a\nsingle electron, so\nthat the change in\nmass can be\nneglected (me �\nmµ+ ,mp,mHe).\n\ninelastic collisions kinematics In an inelastic collision a part of\nthe projectile kinetic energy is lost due to the change of the internal energy\nof the target particle and/or projectile. If Q is the total change of the internal\nenergy in the collision, the energy conservation in the laboratory frame reads\n(assuming vB = 0):\n\n1\n\n2\nmAv\n\n2\nA =\n\n1\n\n2\nmAv\n\n′2\nA +\n\n1\n\n2\nmBv\n\n′2\nB +Q. (47)\n\nThe total kinetic energy is not conserved anymore, while the momentum\nconservation (assuming again vB = 0) takes the same form as for elastic\ncollisions:\n\nmAvA = mAv ′A +mBv ′B. (48)\n\nIn the center-of-mass frame, the total center-of-mass kinetic energy ECM is\nchanged by an amount Q:\n\nE ′CM = ECM −Q. (49)\n\nThis gives us the limit on the maximum possible inelastic energy loss (when\nE ′CM = 0):\n\nQmax = ECM =\nmB\n\nmA +mB\nEA. (50)\n\nFor an inelastic collision, the total energy loss of the projectile in the labora-\ntory frame is the sum of the elastic (T ) and the inelastic (Q) energy losses:\n\n∆E =\n1\n\n2\nmBv\n\n′2\nB +Q (51)\n\n= T +Q. (52)\n\nIn contrast to the elastic-collision case, for inelastic collisions T depends also\non Q and not just on the deflection angle χ. Indeed, the elastic energy loss\nin the laboratory frame reads [25]:\n\nT =\nmAmB\n\n(mA +mB)\n2\nEA\n\n(\n2−Q/ECM − 2\n\n√\n1−Q/ECM cosχ\n\n)\n. (53)\n\nSimilarly, also the scattering angle in the laboratory frame becomes Q-depe-\nndent [25]:\n\ntan ϑA =\nsinχ\n\ncosχ+ mA\n\nmB\n\n1√\n1−Q/ECM\n\n. (54)\n\nNote that, for the “high-energy” collisions, where Q\nECM\n\n� 1, the expressions\nfor T and tan ϑA simplify to the elastic case of Eqs. (43) and (44). This ap-\nproximation is applicable for the simulation of the muon drift in the muCool\ndevice, since the lowest energies at which the inelastic muon-He collisions\n(charge-exchange) are important are around 100 eV, while the typical inelas-\ntic energy loss at these energies is about 10 eV.\n\n\n\n22 theory of muon-helium collisions\n\nUntil now, we have only considered the asymptotic aspects of the collision:\na long time before and after the collision. In this way we have obtained the\nexpressions for the energy loss T and the scattering angle ϑA of the projectile\nin the laboratory system for elastic and inelastic collisions, as a function of\nthe scattering angle χ in the center-of-mass frame (Eqs. (45), (43), (53) and\n(54)). As the deflection angle χ depends on the interaction between the\nparticles during the collision, as a next step we need to consider in more\ndetail what happens to the particles during the collision.\n\nreduction to one particle problem During the collision, the par-\nticles interact via Coulomb interaction, described by a potential V(r), which\ndepends only on the relative distance r between two particles. The force\nresulting from this potential is always pointing along the line connecting\nthe two particles. This implies angular momentum conservation during the\ncollisions. However, the interaction between the two particles via the po-\ntential V(r) changes the linear momentum of the particles, as described by\nNewton’s second law:\n\ndpA\ndt\n\n= −\ndV(r)\n\ndr\nr̂ (55)\n\ndpB\ndt\n\n=\ndV(r)\n\ndr\nr̂, (56)\n\nwhere the radial unit vector r̂ is pointing from particle A to particle B. Ex-\npressing the momenta in the laboratory frame (pA, pB) using center-of-mass\nframe variables:\n\npA = mAvA = mAvC + pCM (57)\n\npB = mBvB = mBvC − pCM, (58)\n\nand inserting them into Eq. (55), we find the following equation of motion\nin the center-of-mass frame:\n\ndpCM\ndt\n\n= −\ndV(r)\n\ndr\nr̂ ⇒ mr\n\ndvr\ndt\n\n= −\ndV(r)\n\ndr\nr̂. (59)\n\nHere we have taken into account that the velocity of the center-of-mass vC\nis constant in time (dvC\n\ndt = 0). Equation (59) tells us that the dynamics of the\ncollision between two particles in the center-of-mass frame, interacting via a\npotential V(r), is the same as for a single particle of mass mr and velocity vr\nmoving in the central potential V(r). This is valid for elastic collisions and\nfor high-energy inelastic ( Q\n\nECM\n� 1) collisions. In the following, we will use\n\nthis finding to simplify the treatment of the collisions.\n\ndeflection function To determine the deflection angle χ in the center-\nof-mass frame we only need to determine the trajectory of the particle with\nmass mr = mAmB\n\nmA+mB\nand initial velocity vr = vA − vB moving in the cen-\n\ntral potential V(r). Since the potential only depends on r, we can write the\nenergy conservation law using polar coordinates (r, φ):\n\nECM =\nmrṙ\n\n2\n\n2\n+\n\nL2\n\n2mrr2\n+ V(r), (60)\n\n\n\n2.2 classical scattering theory 23\n\nwhere L is the angular momentum. As the angular momentum L is con-\nserved during the collision, it can be related to the impact parameter b, i. e.,\nthe distance of closest approach two colliding particles would have if there\nwere no interaction between them (see Fig. 7):\n\nL = mrr\n2φ̇ = mrvrb. (61)\n\nInserting this expression into Eq. (60) we get:\n\nECM =\nmrṙ\n\n2\n\n2\n+\nmrv\n\n2\nrb\n2\n\n2r2\n+ V(r) (62)\n\n=\nmrṙ\n\n2\n\n2\n+ ECM\n\nb2\n\nr2\n+ V(r). (63)\n\nEquation (63) describes a one-dimensional motion of a particle in a so-called\neffective potential Veff, defined as:\n\nVeff ≡ V(r) + ECM\nb2\n\nr2\n. (64)\n\nThe term ECM\nb2\n\nr2\nis often called the centrifugal barrier. The trajectory in the\n\npolar coordinates is described by dr\ndφ :\n\ndr\n\ndφ\n=\nṙ\n\nφ̇\n=\n1\n\nvr\n\nr2\n\nb\n\ndr\n\ndt\n, (65)\n\nwhere we used Eq. (61) to express φ̇ as a function of b: φ̇ = vr\nb\nr2\n\n. The\nderivative ṙ = dr\n\ndt can be obtained for all times t (at all r) from Eq. (63):\n\ndr\n\ndt\n=\n\n(\n2\n\nmr\n\n) 1\n2\n(\nECM − V(r) − ECM\n\nb2\n\nr2\n\n)\n(66)\n\n= vr\n\n(\n1−\n\nV(r)\n\nECM\n−\nb2\n\nr2\n\n)\n, (67)\n\nwhich means that the trajectory of the particle is determined by the following\nexpression (obtained by combining Eq. (67) with Eq. (65)):\n\ndr\n\ndφ\n=\nr2\n\nb\n\n(\n1−\n\nV(r)\n\nECM\n−\nb2\n\nr2\n\n)\n. (68)\n\nWe can see that the trajectory is uniquely determined by the center-of-mass\nenergy ECM and by the impact parameter b.\n\nThe deflection angle χ in the center-of-mass frame can now be determined\nfrom the particle trajectory equation:\n\nχ = π− 2φ0 = π− 2\n\nφ0∫\n0\n\ndφ = π− 2\n\n∞∫\nr0\n\n1\n\ndr/dφ\ndr, (69)\n\nwhere r0 is the point of closest approach (also called the turning point) and\nφ0 and χ are defined in Fig. 7 (bottom). The total deflection of the particle\ndepends on how close the particle comes to the center of force, i.e. on r0.\n\n\n\n24 theory of muon-helium collisions\n\nIntuitively, the closer two particles come during the collision, the more is\nthe initial trajectory of the projectile modified (larger deflection angle). The\npoint of the closest approach r0 to the center of force (r = 0) can be calculated\nusing the condition:\n\ndr\n\ndt\n\n∣∣∣∣\nr0\n\n= 0, (70)\n\nwhich combined with Eq. (67) gives us the following result:\n\nr20 =\nb2\n\n1− V(r0)/ECM\n. (71)\n\nFor repulsive potentials, the distance of the closest approach is larger than\nb, while for attractive potentials it is smaller than b. If the center-of-mass en-\nergy is relatively large compared to the interaction potential (V(r0)/ECM �\n1), the point of the closest approach is roughly equal to the impact parameter\n(r0 ≈ b), implying only small deflections of the projectile.\n\nThe exact deflection angle χ is obtained by combining Eq. (69) with Eq. (68):\n\nχ(b) = π− 2b\n\n∫∞\nr0\n\ndr\n\nr2\n1√\n\n1−\nV(r)\nECM\n\n− (br )\n2\n\n. (72)\n\nThis formula allows us to calculate the center-of-mass scattering angle χ\nas a function of the impact parameter b, which is often referred to as the\ndeflection function χ(b). The deflection function clearly also depends on the\nECM and the shape of the interaction potential V(r).\n\nIn general, the deflection angle is always positive for purely repulsive\npotentials, and negative for purely attractive potentials (see Fig. 8). For large\nimpact parameters b, the deflection angle χ is small (for any potential that\ngoes to 0 as r → ∞). For small impact parameters, the deflection angle is\nlarge for potentials that are repulsive at small r. For such potentials, the\nlargest deflection (χ = π) occurs in a head-on collision, i.e., for b = 0.\n\nFurthermore, we can note that the largest contribution to the integral in\nEq. (72) comes from the part of the effective potential close to the turning\npoint r0, where the denominator is the smallest. Physically, this is because\nthe particle radial velocity is the smallest at this point, so that the particle\nspends most of the time during the collision in the region around the turning\npoint. Thus, the angle χ often depends mainly on the shape of the interaction\npotential close to the turning point.\n\nIntuitively, the steeper the potential around the turning point is, the larger\nis the deflection angle χ, because the particles experience a larger force\n(which is the derivative of the potential). This can be seen most easily for\nhigh energies, when V(r)/ECM � 1 and thus r0 ≈ b, by expanding the\ndenominator of Eq. (72) in a Taylor series around r0 [26]:\n\nχ(b) ≈ −\nb\n\nECM\n\n∞∫\nb\n\ndV(r)\n\ndr\n\ndr√\nr2 − b2\n\n. (73)\n\n\n\n2.2 classical scattering theory 25\n\nFigure 8: Sketch of the deflection function χ(b) for purely attractive potential V(r) ∝\n−1/r (left) and for purely repulsive potential V(r) ∝ 1/r (right). The\ntrajectories in the collision plane for several impact parameters b are also\ngiven to relate the shape of the deflection function to the particle motion\nduring the collision. For simplicity, the deflection angle is indicated only\nfor two trajectories.\n\nIn this case, the deflection angle directly depends on the derivative of the\npotential dVdr and it is inversely proportional to the center-of-mass energy (at\na fixed impact parameter b):\n\nχ ∝ 1\n\nECM\n. (74)\n\nFor power-law potentials, V(r) = C/rn, where the constant C can be positive\nor negative, Eq. (73) can be evaluated analytically, leading to the following\nform of the deflection function for the small angles [27]:\n\nχ(b) ∝ 1\n\nECM\n\n1\n\nbn\n. (75)\n\nFor more complex potentials, the deflection function usually cannot be eval-\nuated analytically. But often the realistic potential can be approximated with\npower-law potentials close to the turning point. Hence, using Eq. (75), we\n\n\n\n26 theory of muon-helium collisions\n\ncan approximately infer the behavior of the deflection function for any po-\ntential form, at least for small deflection angles.\n\nIn summary, up to now we have obtained an expression (Eq. (72)) that al-\nlows us to calculate the deflection angle χ in the center-of-mass frame, given\nthe initial energy ECM, the impact parameter b and the interaction poten-\ntial V(r). The obtained deflection angle can be used to calculate the elastic\nenergy loss, according to Eqs. (44) and (53), and the laboratory scattering\nangles, according to Eqs. (43) and (54).\n\nAll these quantities depend on the value of the impact parameter, which\nis not a measurable quantity. However, the impact parameter can be directly\nrelated to the collision probability into a certain angle χ, i. e., to the classical\nangular differential cross section, as we will see in the next subsection.\n\nclassical cross section from the deflection function\n\nThe goal of this section is to obtain the distribution of χ, namely the angular\ndifferential cross section dσ\n\ndΩ\n(χ,ECM), for the elastic collisions. To simplify\n\nthe notation, we define:\n\nIEL (χ,ECM) ≡ dσ\n\ndΩ\n(χ,ECM) . (76)\n\nSince the interaction between the colliding particles that we are considering\nin this chapter depends only on the radial distance r, the scattering should\nbe isotropic in azimuthal angle. Therefore, in our considerations, the angular\ndifferential cross section will be a function of the polar angle χ only.\n\nThe number of particles scattering elastically in the solid angle between\nΩ and Ω+ dΩ per unit time, per unit flux of the incident particles and per\ntarget atom is:\n\ndσEL (χ,ECM) = IEL (χ,ECM)dΩ = 2π IEL (χ,ECM) sinχdχ. (77)\n\nThe angular distribution can be obtained from the deflection function χ(b)\nof Eq. (72) by remarking that only the particles with the impact parameter\nbetween b and db scatter into the angle between χ(b) and χ(b) + dχ(b), i. e.,\nthe angle χ is a one-to-one function of b. The number of particles having\nthe impact parameter between b and db is proportional to 2πbdb and thus,\nfrom Eq. (77) we get [27]:If there were\n\nseveral impact\nparameters b\n\nproducing the same\ndeflection χ,\n\nclassically their\ncontributions\n\nwould simply be\nsummed.\n\nIEL (χ,ECM) =\nb\n\nsinχ\n\n∣∣∣∣dbdχ\n∣∣∣∣ (78)\n\n=\nb\n\nsinχ\n1\n\n|dχ(b)/db|\n. (79)\n\nWe can notice that the scattering probability increases with b. As we have\nseen before, large impact parameters in general result in small deflections, so\nthat the differential cross section increases with decreasing angles (forward-\npeaked shape).\n\n\n\n2.2 classical scattering theory 27\n\nThis forward-peaking can be demonstrated directly for power-law poten-\ntials of the form V(r) = Cr−n. By using Eq. (75) to express b(χ) for the small\ndeflections χ and inserting it into Eq. (78) we find that:\n\nIEL (χ,ECM) ∝ 1\n\nE\n2/n\nCM\n\n1\n\nχ1+2/n sinχ\n, (80)\n\nwhich confirms the previous conclusion that the differential cross sections\nare in general strongly forward peaked.\n\nThe total elastic cross section at an energy ECM is obtained by integrating\nIEL (χ,ECM) over all angles:\n\nσEL (ECM) = 2π\n\nπ∫\n0\n\nIEL (χ,ECM) sinχdχ. (81)\n\nBy inserting the expression for IEL from Eq. (78) into Eq. (81), we can see\nthat the expression above is equivalent to the integral over all possible impact\nparameters:\n\nσEL (ECM) = 2π\n\nb(χ=0)∫\nb(χ=π)\n\nbdb. (82)\n\nFor inelastic collisions, the internal state of the particles involved in the col-\nlisions is changed, i. e., they involve a transition from one internal (initial)\nstate to another (final) state. This transition can happen with some prob-\nability P, which can depend on the impact parameter b and on the initial\nvelocity/energy (the exact dependencies and values have to be evaluated\nquantum mechanically). Thus we need to modify the expressions for the\ndifferential cross sections accordingly:\n\nIINEL (χ,ECM) = P(b,ECM) · b\n\nsinχ\n\n∣∣∣∣dbdχ\n∣∣∣∣ (83)\n\n= P(χ,ECM) · IEL (χ,ECM) , (84)\n\nwhere in the second line we have inserted Eq. (78). The total cross section is\nthen obtained by integrating over all solid angles:\n\nσINEL (ECM) = 2π\n\nπ∫\n0\n\nIINEL (χ,ECM) sinχdχ (85)\n\n= 2π\n\nπ∫\n0\n\nP(χ,ECM) · IEL (χ,ECM) sinχdχ (86)\n\n= 2π\n\nb(χ=0)∫\nb(χ=π)\n\nP(b,ECM) · bdb. (87)\n\n\n\n28 theory of muon-helium collisions\n\nstopping power We are now in the position to determine the stopping\npower for elastic and inelastic collisions. For purely elastic collisions, the\nenergy loss is equal to the elastic energy loss ∆E = T (Q = 0). Hence, using\nEq. (25), we obtain (for purely elastic collisions):\n\nSEL(ECM) =\n\n∫\nT(χ,ECM)dσEL(χ,ECM) (88)\n\n= 2π\n\nπ∫\n0\n\nT(χ,ECM)IEL (χ,ECM) sinχdχ (89)\n\n= 2\nmA\n\nmA +mB\nECM · 2π\n\nπ∫\n0\n\n(1− cosχ) IEL (χ,ECM) sinχdχ,\n\n(90)\n\nwhere we used Eqs. (45) and (77) to express T and dσEL. The last integral is\nthe so-called diffusion or momentum-transfer cross section:\n\nσMT (ECM) ≡ 2π\nπ∫\n0\n\nIEL (χ,ECM) (1− cosχ) sinχdχ. (91)\n\nThe momentum transfer cross section is determined primarily by the differ-\nential cross section IEL at large deflection angles χ, i.e., at deflection angles\nthat produce significant energy losses.\nUsing the momentum-transfer cross section, the elastic stopping power of\nEq. (90) can be rewritten as:\n\nSEL(ECM) = 2\nmA\n\nmA +mB\nECM · σMT (ECM). (92)\n\nIn a similar way we can define the stopping power for inelastic collisions,\nwhere the total energy loss is the sum of the elastic and the inelastic energy\nlosses (∆E = T +Q):\n\nSINEL(ECM) =\n\n∫\n[T(χ,Q,ECM) +Q]dσINEL(χ,Q,ECM) (93)\n\n=\n\n∫\nQ · dσINEL(χ,Q,ECM)\n\n+\n\n∫\nT(χ,Q,ECM) · dσINEL(χ,Q,ECM) (94)\n\n= Q · σINEL(ECM)\n\n+\n\n∫\nT(χ,Q,ECM) · dσINEL(χ,Q,ECM),\n\nwhere Q is the average inelastic energy loss at a certain energy ECM, and T\nis given by Eq. (53). For “high-energy” inelastic collisions, where Q� ECM,\nthe elastic energy loss T is almost the same as in purely elastic collisions, as\ndiscussed already before. In this limit, the stopping power can be written as\n\nSINEL(ECM) = Q · σINEL(ECM) (95)\n\n+ 2\nmA\n\nmA +mB\nECM · 2π\n\nπ∫\n0\n\n(1− cosχ) IINEL (χ,ECM) sinχdχ.\n\n\n\n2.3 high-energy regime 29\n\nSince the angular differential cross section IINEL is typically forward-peaked, The factor of\n1− cosχ in the\nintegral of Eq. (95)\nsuppresses the\nsmall-angle\ncontributions.\n\nthe integral of Eq. (95) is much smaller than the total inelastic cross section\nσINEL and can be neglected. Hence, the inelastic stopping power simplifies\nto:\n\nSINEL(ECM) ≈ Q · σINEL(ECM). (96)\n\n2.3 high-energy regime\n\nIn this section we will apply the classical scattering theory to muon-He\n(proton-He) collisions at “high” energy, where the energy loss is dominated\nby inelastic processes leading to the excitation and the ionization of the tar-\nget atoms (see Eqs. (17) and (18)). In this regime we can assume that the\nprojectile is colliding only with a single target electron or nucleus, but not\nwith the whole atom. This is known in the literature as the binary encounter\napproximation [25]. For the scattering off the target electrons, we also make\nan additional approximation, the so-called impulse approximation [25]. In\nthis approximation, in a first step the binding of the electrons is ignored\nduring the collision. The scattering angle and the energy loss are then com-\nputed assuming an elastic collision with the electron. After the collision, the\nbinding energy of the electron is “turned on”, so that the energy loss com-\nputed in the previous step is (partially) used to excite or ionize the target\natom. Since the electronic levels are discrete (due to quantization), this will\nlimit the range of the possible elastic energy transfers.\n\nCollisions at velocities above the Bohr velocity First we derive the stop-\nping power for energies where the projectile velocity is much higher than\nthe Bohr velocity of the electron of the target atom. For muons in He, this\ncorresponds approximately to energies much higher than 5 keV, while for\nprotons this regime is applicable for energies higher than 50 keV. In this\nenergy regime, we can consider the target electron to be at rest, since its\nvelocity is much lower than the projectile velocity. In conjunction with all\npreviously mentioned assumptions, this means that we treat the collisions\nin this energy regime as collisions with free electron or free nucleus at rest.\n\nIn that case the interaction potential is simply:\n\nV(r) =\nqAqB\nr\n\n, (97)\n\nwhere qA and qB are the projectile and target particle charges, respectively.\nTo determine the stopping power, we need to determine the differential cross\nsection for this potential. For this purpose, we first need to find the deflec-\ntion function from Eq. (72). Using the potential of Eq. (97), the deflection\nfunction can be evaluated analytically, giving [28]:\n\nχ(b) = 2 tan−1\n\n(\nqAqB\n2ECMb\n\n)\n. (98)\n\nInverting the equation above to obtain b(χ) we get:\n\nb(χ) =\nqAqB\n\n2ECM tan(χ/2)\n, (99)\n\n\n\n30 theory of muon-helium collisions\n\nwhich we can insert into Eq. (78) to obtain the angular differential cross\nsection IEL(χ,ECM)\n\nIEL (χ, ECM) =\nq2Aq\n\n2\nB\n\n8E2CM sin4(χ/2)\n. (100)\n\nThis is the well-known Rutherford cross section. We can see that the an-\ngular distribution is strongly peaked at small angles χ, which is a general\nfeature of the scattering on power-law potentials, as already discussed in the\nprevious section.\n\nStopping power After having obtained the differential cross section\nIEL (χ, ECM), we can calculate the stopping power due to collisions with\neither electrons or nuclei, using Eq. (90)2:\n\nS(ECM) = 2π\n\nπ∫\n0\n\nT(χ,ECM)IEL (χ,ECM) sinχdχ (101)\n\n= 2\nmA\n\nmA +mB\nECM · 2π\n\nπ∫\n0\n\n(1− cosχ) IEL (χ,ECM) sinχdχ\n\n(102)\n\n= 2\nmA\n\nmA +mB\nECM ·\n\nq2Aq\n2\nB\n\n8E2CM\n· 2π\n\nπ∫\n0\n\n(1− cosχ)\nsin4(χ/2)\n\nsinχdχ (103)\n\n= 2\nmA\n\nmA +mB\n·\nq2Aq\n\n2\nB\n\n8ECM\n· 8π\n\n1∫\n−1\n\nd (cosχ)\n(1− cosχ)\n\n. (104)\n\nThe last integral diverges as a consequence of (temporarily) ignoring the\nfact that the target nucleus and electrons are not free (they are bound in\nthe atom). Indeed, as the Coulomb potential has infinite range, arbitrarily\nsmall deflections (and thus energy transfers) are possible, which causes the\ndivergence of the differential cross section at small deflection angles.\n\nBut in reality, a cut-off for the lowest allowed deflection χmin exists, owing\nto the fact that target electrons and nuclei are not actually free. The cut-off\n\n2 We use the equation for the stopping power derived for the elastic collisions because we have\nassumed that the scattering at these energies behaves as if the muon (proton) was scattering\noff a free electron or nucleus.\n\n\n\n2.3 high-energy regime 31\n\nangle χmin can then be converted into a energy transfer cut-off Tmin, using\nEq. (45). Thus, we can write the stopping power as:\n\nS(ECM) = 2π\nmA\n\nmA +mB\n·\nq2Aq\n\n2\nB\n\nECM\n\ncosχmin∫\n−1\n\nd (cosχ)\n(1− cosχ)\n\n(105)\n\n= 2π\nmA\n\nmA +mB\n·\nq2Aq\n\n2\nB\n\nECM\nln\n(\n\n2\n\n1− cosχmin\n\n)\n(106)\n\n= 2π\nmA\n\nmA +mB\n·\nq2Aq\n\n2\nB\n\nECM\nln\n(\nTmax\n\nTmin\n\n)\n(107)\n\n= 4π\nq2Aq\n\n2\nB\n\nmBv\n2\nA\n\nln\n(\nTmax\n\nTmin\n\n)\n. (108)\n\nIn the third step we have used Eqs. (45) and (46) to express the argument\nof the logarithm in terms of minimum and maximum elastic energy loss,\nTmin and Tmax, corresponding to deflection angles χmin and χmax = π.\nUsing Eq. (108) we can estimate the stopping power due to the scattering off\nelectrons and nuclei separately.\n\nThe stopping power per target atom due to proton (muon) collisions with\nelectrons (electronic stopping) is:\n\nSe = 4πZB ·\ne4\n\nmev\n2\nA\n\nln\n(\nTmax,e\n\nTmin,e\n\n)\n, (109)\n\nwhere me is the electron mass and |qA| = |qB| = e are the proton (muon)\nand electron charges. The additional factor ZB (atomic number) is inserted\nto account for incoherent scattering off all the electrons of the atom (the\nstopping power is defined per atom).\n\nFrom Eq. (46) we can obtain a simple expression for the maximal energy\ntransfer to an electron in an elastic collision: Tmax,e ≈ 2mev\n\n2\nA, since the\n\ntarget particle mass (me) is much smaller than the projectile mass mA (mµ,\nmp). For the scattering off bound electrons, the cut-off Tmin,e is a conse-\nquence of the electron binding. The minimal energy transfer to an electron\nin the first approximation corresponds to the lowest possible excitation of the\ntarget atom: Tmin,e ≈ Qmin. Thus, the electronic stopping power becomes:\n\nSe = 4πZB ·\ne4\n\nmev\n2\nA\n\nln\n(\n2mev\n\n2\nA\n\nQmin\n\n)\n. (110)\n\nNote that the stopping power depends only on the velocity vA of the projec-\ntile, but not on its mass mA. Therefore, the electronic stopping power for\nmuons in He is same as the electronic stopping power for protons in He at\nthe same velocity (velocity scaling):\n\nSe,p(vA) = Se,µ(vA) , (111)\n\nwhere the indices p and µ refer to protons and muons, respectively.\n\n\n\n32 theory of muon-helium collisions\n\nMoving on to consider the stopping power due to muon (proton) collisions\nwith the target nuclei (nuclear stopping power), we obtain the following\nexpression:\n\nSn = 4π\nZ2Be\n\n4\n\nmBv\n2\nA\n\nln\n(\nTmax,n\n\nTmin,n\n\n)\n, (112)\n\nwhere we inserted for the nuclear charge qB = ZBe. In our case, mB is the\nHe mass and ZB = 2.\n\nThe Tmax,n is again given by Eq. (46): Tmax,n = 2\nm2\nAmB\n\n(mA+mB)\n2 v\n2\nA. Regarding\n\nTmin,n, for the collision with nuclei, the physical origin of the cut-off is\nthe screening of the target nucleus by the orbiting electrons. Due to this\nscreening effect, the interaction potential does not have an infinite range\nanymore. Hence, only impact parameters smaller than the potential range\na contribute to the stopping power. The lowest possible deflection can be\nobtained from Eq. (98), by setting b = a:\n\nχmin = 2 tan−1\n\n(\nqAqB\n2ECMa\n\n)\n, (113)\n\nyielding, from Eq. (44), a minimum allowed energy transfer of\n\nTmin = 2\nmAmB\n\n(mA +mB)\n2\nEA (1− cosχmin) (114)\n\n= 4\nmAmB\n\n(mA +mB)\n2\nEA\n\n1\n\n1+\n(\n2ECMa\nqAqB\n\n)2 (115)\n\n= Tmax\n1\n\n1+\n(\n2ECMa\nqAqB\n\n)2 . (116)\n\nInserting this into the expression for the stopping power of Eq. (112), we get:\n\nSn = 4π\nZ2Be\n\n4\n\nmBv\n2\nA\n\nln\n\n[\n1+\n\n(\n2ECMa\n\nqAqB\n\n)2]\n(117)\n\n≈ 4π\nZ2Be\n\n4\n\nmBv\n2\nA\n\nln\n(\n2ECMa\n\nqAqB\n\n)2\n(118)\n\n= 4π\nmA\n\nmA +mB\n·\nZ2Be\n\n4\n\nECM\nln\n(\n2ECMa\n\nqAqB\n\n)2\n, (119)\n\nwhere mB is in this case the He mass. From the last equation it follows\nthat, at the same center-of-mass energy ECM, the nuclear stopping power\nfor protons in helium is a factor of mp\n\nmp+mHe\n· mµ+mHe\n\nmµ\n≈ 7.3 larger than the\n\nnuclear stopping power for muons in helium:\n\nSn,p(ECM) = 7.3 · Sn,µ(ECM). (120)\n\nIt is also interesting at this point to compare the electronic and the nuclear\nstopping powers:\n\nSn\n\nSe\n= ZB\n\nme\n\nmB\nln\n(\n2ECMa\n\nqAqB\n\n)2\n/ ln\n\n(\n2mev\n\n2\nA\n\nQmin\n\n)\n. (121)\n\n\n\n2.3 high-energy regime 33\n\nSince the electron mass me is more than 3 orders of magnitude smaller than\nthe nucleus mass, the nuclear stopping power at these “high” energies is\nmuch smaller than the electronic stopping power. The ratio of the logarithms\nin the equation above cannot compensate3 for the difference caused by the\nmass ratio me/mB.\n\nCollisions at velocities below the Bohr velocity Let us now consider\nbriefly what happens when the projectile velocity becomes smaller then the\norbital velocity of the target electron, i. e., when the muon (proton) energy\nis smaller than 5 keV (50 keV). In this regime, the electron motion cannot\nbe ignored anymore, and the electronic stopping power given by Eq. (110)\nhas to be modified. It is commonly assumed that in this case the electronic\nstopping power is linearly proportional to projectile velocity [29]:\n\nSe ∝ vA. (122)\n\nThe important feature is that the stopping power still depends only on the\nprojectile velocity vA and not on its massmA. Thus, the previous conclusion\n- that the electronic stopping power for muons in He is the same as the\nelectronic stopping power for protons in He at the same velocity (velocity-\nscaling of the stopping power) - is still valid.\n\nComparison with SRIM As a summary, in Fig. 9 we show the total stop-\nping power (black solid line) for protons in He as a function of the proton\nenergy (in the laboratory reference frame), obtained from the SRIM soft-\nware [30]. The contributions to the total stopping power from nuclear and\nelectronic stopping are shown separately (blue and red dashed curves). As\ndemonstrated before, for “high-energy” collisions (Ep > 10 keV) the nuclear\nstopping power is several orders of magnitude smaller than the electronic\nstopping power. As detailed above, we can identify two distinct regions\nof stopping power behavior in the “high-energy” regime. The first region\ncorresponds to the stopping power when the projectile velocity is higher\nthan the Bohr velocity. Using several assumptions we showed that, in this\nregime, the stopping power decreases with increasing velocity: Se ∝ v−2A\n(see Eq. (110)). The second region, where the projectile velocity is lower\n\n3 We can estimate the ratio of the logarithms if we assume that the range of the potential is\napproximately given by the Bohr radius (a = a0 = 1 a.u.). The physical interpretation of\nthis assumption is that projectile needs to penetrate the electron cloud in order to \"see\" the\nnucleus. If the target atom is He, the minimum inelastic energy loss Qmin is approximately\ngiven by the lowest excitation energy of He, Qmin = 18.7 eV . For protons colliding with He,\nthe ratio of logarithms of Eq. (121) is:\n\n2 ln\n(\nECM\n27.2 eV\n\n)\n/ ln\n\n(\n2mev\n\n2\nA\n\n18.7 eV\n\n)\n= 2 ln\n\n(\n0.8EA\n27.2 eV\n\n)\n/ ln\n\n(\n4\n\n1836\n\nEA\n18.7 eV\n\n)\nwhere we have inserted mA = mp = 1836me and mB = 4mp, with mp being the proton\nmass. For EA = 50 keV the ratio of logarithms evaluates to about 8, while for EA = 4 MeV\n\nthe ratio is about 4, which is negligible compared to the factor me/mB ≈ 1\n4·1836 in Eq. (121).\n\nFor muons colliding with He, the results are very similar.\n\n\n\n34 theory of muon-helium collisions\n\nthan the Bohr velocity, has a different behavior: the stopping power here in-\ncreases with the increasing velocity. This is consistent with the dependence\nof Eq. (122), where Se ∝ vA. In-between those two regions, the stopping\npower goes through a maximum: the so-called Bragg peak.\n\nHowever, in the same plot, we can notice that below 10 keV proton en-\nergy (indicated by the “threshold” line), the electronic stopping power drops\nmore rapidly than linearly with the velocity vA. At this energy, the maxi-\nmum kinematically allowed energy transfer to the electron approaches the\nlowest possible excitation energy of the He atom. This is known in litera-\nture as the “threshold effect”, and it has been confirmed both theoretically\nand experimentally for protons in helium gas [29, 31–33]. As a consequence\nof this decrease of the electronic (inelastic) stopping power with decreas-\ning projectile energy, the elastic (nuclear) stopping power will eventually\nbecome larger than the inelastic (electronic) stopping power. For protons in\nHe, this happens at around 1 keV energy. Indeed, the nuclear (elastic) stop-\nping power has no such threshold as the projectile can always lose energy in\nan elastic collision with a target particle at rest.\n\nWe can conclude that, for protons in He, the threshold energy of 10 keV\nrepresents the lower limit of the “high-energy” stopping regime, where the\nexcitation and ionization of the target atoms play the dominant role in the\nslowing down of the projectile. Below these energies, other proton-He inter-\nactions become responsible for the energy loss, mainly charge-exchange and\neventually, at even lower energies, the elastic collisions with target atoms.\nThese processes will be the focus of the following section.\n\n2.4 low-energy regime\n\nAt low energy (Ep < 10 keV, Eµ < 1 keV), the assumption used earlier, that\nthe projectile collides only with a single electron or a nucleus is not war-\nranted anymore. Since the projectile is relatively slow, it spends a long time\n(compared to the period of the orbiting electron) in the vicinity of the He\natom, so that it interacts both with the electrons and the nucleus. Since the\nvelocity of the projectile is much lower than that of the target electrons in\nthis energy regime, during the collision a quasi-molecular ion is formed, con-\nsisting of the projectile and the target nuclei surrounded by the much faster\ntarget electrons. The interaction potential between the projectile and the tar-\nget atom can then be calculated in the Born-Oppenheimer approximation.\n\nBecause the target electrons are basically shared between projectile and\ntarget nuclei in the quasi-molecule formed during the collision, it becomes\nlikely for a target electron to end up bound to the projectile after the colli-\nsion. Thus, the charge exchange process becomes an important energy loss\nmechanism in this energy regime.\n\nLow-energy protons colliding with helium atoms form HeH+ molecular\nions. The potential energy curves of the ground state and several excited\nstates of the HeH+ molecular ion are shown in Fig. 10. Each of the molecular\nstates has a different electron arrangement, which correlates with different\nproton and helium states in the separated atoms limit (r→∞). For example,\n\n\n\n2.4 low-energy regime 35\n\nFigure 9: The electronic (Se), nuclear (Sn) and total (Se + Sn) stopping powers for\nprotons in He as a function of the proton energy (in the laboratory refer-\nence frame), obtained from the SRIM software [30]. The various energy\nregimes, discussed in this chapter, are also sketched: the ”high-energy”\nregime (above 10 keV proton energy) and the low-energy regime (below\n10 keV proton energy). In the “high-energy” regime, two distinct behav-\niors of the electronic stopping power can be discerned, depending on\nwhether the proton velocity vA is higher (brown region) or lower (red\nregion) than the Bohr velocity vBohr of the electron in the He atom. In the\nlow-energy regime, again two energy ranges can be distinguished: where\ninelastic (electronic) processes dominate the stopping power (orange re-\ngion) and where the elastic (nuclear) scattering dominates the stopping\npower (blue region).\n\n\n\n36 theory of muon-helium collisions\n\nthe HeH+ ground state (1Σ) corresponds to an asymptotic configuration\nwith a free He atom in the ground state (1s2) and a free proton, while the first\nexcited state (2Σ) corresponds to a configuration where the quasi-molecule\nseparates into a hydrogen atom and He+ ion, both in their ground states\n(1s).\n\nThe potential energy curves of Fig. 10 are the so-called adiabatic potential\nenergy curves, i. e., they were calculated assuming stationary nuclei. During\nthe collision, the nuclei participating in the collision however do move (al-\nthough slowly compared to the electrons). This nuclear motion perturbs the\nelectron motion, which can result in a transition into a different state of the\nmolecular ion, corresponding to an inelastic collision in the separated atoms\nlimit. If, for example, the transition occurs from the HeH+ ground state 1Σ\nto the first excited state 2Σ, the collision corresponds to an electron-capture\ncollision, as described by Eq. (19). On the other hand, if no transition to\nanother molecular state happens, the collision is elastic.\n\nThe knowledge of the interaction potentials (Fig. 10 (top)), allows us to\ncalculate the cross sections and stopping powers for the elastic and inelastic\nprocesses using the procedure outlined in Section 2.2, in a similar way as\nwas done in Section 2.3 for the “high-energy” regime. However, in this case,\nthe interaction potential is not of a simple form, where the deflection func-\ntion can be calculated analytically from Eq. (72): a numerical calculation is\nneeded. But, as detailed below, the scaling from proton to muon cross sec-\ntions and stopping powers can already be deduced by some simple physical\narguments, without performing the complete calculation.\n\nFurthermore, already by considering how the shape of the interaction po-\ntential would affect the muon (proton) trajectories in the low-energy colli-\nsions, several conclusions may be drawn about the distribution of the deflec-\ntion angle χ, i. e., about the shape of the angular differential cross sections.\nThis is important for estimating the range of validity of the classical scatter-\ning picture.\n\nTo begin with, we will study the behavior of the low-energy inelastic col-\nlisions on the example of electron capture, which is one of the dominant\ninelastic processes at energies in the energy range from 1 keV to 10 keV for\nprotons and 100 eV to 1 keV for muons. Then, we will move on to examine\nthe low-energy elastic collisions, which become important at energies below\n1 keV (100 eV) for protons (muons).\n\nelectron capture The following part of this section describes the de-\ntails of the electron-capture (EC) collision on the example of the electron\ncapture by the proton from a helium atom in the ground state, forming a\nhydrogen atom in the ground state:\n\nH+ +He(1s2)→ H(1s) +He+(1s). (123)\n\nAs explained earlier, such electron capture happens only when a transition\nfrom the ground state (1Σ) to the excited state (2Σ) of the HeH+ molecular\nion occurs. The inelastic energy loss Q is then determined by the difference\nin the internal energy between the initial and the final states at r → ∞. In\n\n\n\n2.4 low-energy regime 37\n\nFigure 10: (Top) Adiabatic potential energy curves of various states of the HeH+\n\nmolecular ion. Reproduced from [34]. (Bottom) Radial coupling between\nthe ground state 1Σ and the first excited state (2Σ) of the HeH+ molecular\nion. Reproduced from [35].\n\n\n\n38 theory of muon-helium collisions\n\nthis case, the inelastic energy loss is given by the difference between the ion-\nization energies of hydrogen and helium atoms: QEC = 24.6 eV − 13.6 eV =\n\n11 eV4.\nThe transition between the 1Σ and the 2Σ states in HeH+ is possible be-\n\ncause these two states are coupled non-adiabatically. In other words, there\nis a coupling between the two states caused by the perturbation associated\nwith the motion of the nuclei (H+ and He+ in this case) during the collision.\nIn this particular example, the two levels are coupled via the so-called radial\ncoupling [35, 36]. The radial coupling between the 1Σ and 2Σ states as a\nfunction of the distance between the two particles is shown in Fig. 10 (bot-\ntom). We can see from this figure that the coupling has a narrow maximum\naround a radial distance r = 2.5 a0 ≡ Rx, which means that in this region,\nthe transition between two states is the most probable.\n\nDuring the collision, the radial separation r between the two nuclei de-\ncreases down to r = r0 (turning point) and then it increases again (see Fig. 7).\nTherefore, the transition region at r ≈ Rx is passed twice during the collision,\nthe first time on the way towards the turning point, and then again on the\nway back from the turning point. In other words, the transition from the 1Σ\nstate to the 2Σ state (equivalent to electron capture) can occur in two ways:\nthe first while the colliding nuclei approach each other, the second when\nthey fly apart, as sketched in Fig. 11.\n\nIn the first way, the colliding particles interact via the excited state po-\ntential (2Σ) for most of the time during the collision, while in the second\nway, the colliding particles spend most of the time during the collision in-\nteracting via the ground state potential 1Σ. As a result, each way yields a\ndistinct deflection function, resulting in two different trajectories of the col-\nliding particles. These two deflection functions are sketched in Fig. 12. In\nboth cases, the scattering angle χ is positive as the collisions are dominated\nby the repulsive core of the potential.\n\nThe deflection functions for the two cases can be calculated using Eq. (72),\nwhile taking care to use different interaction potentials before and after the\ntransition. For the trajectory #1, this reads:\n\nχ1(b) = π+ b\n\n∫Rx\n∞\ndr\n\nr2\n1√\n\n1−\nV1(r)\nECM\n\n− (br )\n2\n\n+ b\n\n∫r0,1\n\nRx\n\ndr\n\nr2\n1√\n\n1−\nV2(r)\nECM\n\n− (br )\n2\n\n− b\n\n∫∞\nr0,1\n\ndr\n\nr2\n1√\n\n1−\nV2(r)\nECM\n\n− (br )\n2\n\n, (124)\n\nwhere V1(r) is the interaction potential of the HeH+ ground state (1Σ) and\nV2(r) is the interaction potential of the HeH+ first excited state (2Σ). In this\n\n4 If the capture were to occur into some higher excited state of the hydrogen atom, the inelastic\nenergy loss would be higher. Hence, such processes are less likely than the capture into the\nground state, given the low energy of the projectile that we are considering here.\n\n\n\n2.4 low-energy regime 39\n\nFigure 11: Sketch of the two possible ways that a transition from the HeH+ ground\nstate (1Σ) to the first excited state (2Σ) can occur. (Top) The transition\noccurs while the colliding particles are approaching. (Bottom) The tran-\nsition occurs when the particles are flying apart. For the top scenario,\nduring the collision the particles interact most of the time via the 2Σ po-\ntential, while for the bottom scenario, they interact most of the time via\nthe 1Σ potential. Inspired by a similar sketch of [25].\n\n\n\n40 theory of muon-helium collisions\n\nFigure 12: Sketch of the two deflection functions χ(b), corresponding to the two\ndifferent ways a transition from the ground state to the excited state of\nthe HeH+ molecular ion can occur (see Fig. 11). This transition is only\npossible up to a maximal impact parameter bmax (see text).\n\ncase, the turning point r0,1 of the first trajectory depends on the potential\nV2(r) via Eq. (71):\n\nr20,1 =\nb2\n\n1− V2(r0,1)/ECM\n. (125)\n\nSimilarly, for the trajectory #2, the deflection function can be written in\nthe following way:\n\nχ2(b) = π+ b\n\n∫r0,2\n\n∞\ndr\n\nr2\n1√\n\n1−\nV1(r)\nECM\n\n− (br )\n2\n\n+ b\n\n∫Rx\nr0,2\n\ndr\n\nr2\n1√\n\n1−\nV1(r)\nECM\n\n− (br )\n2\n\n− b\n\n∫∞\nRx\n\ndr\n\nr2\n1√\n\n1−\nV2(r)\nECM\n\n− (br )\n2\n\n. (126)\n\nThe turning point r0,2 of the trajectory #2, depending on the potential V1(r),\nas:\n\nr20,2 =\nb2\n\n1− V1(r0,2)/ECM\n, (127)\n\nis different from the turning point r0,1 of the trajectory #1, as V1 6= V2.\nIn order to calculate the differential cross section, we need one additional\n\ningredient: the transition probability P between the two states (see Eq. (83)).\nCalculation of such a transition requires in principle a quantum-mechanical\n\n\n\n2.4 low-energy regime 41\n\napproach. However, some simplified arguments about the form (dependen-\ncies) of the transition probability can be made using a classical approach by\nconsidering the trajectories of the colliding nuclei.\n\nEach time the transition region around r = Rx is passed, there is some\nprobability of excitation to the upper state, pexc, if the molecular ion is in\nthe ground state. Likewise, if the molecular ion has already been excited to\nthe upper state in the first part of its trajectory, there is a probability that it\nde-excites back to the ground state the second time it passes the transition\nregion at r = Rx. We assume that the de-excitation probability is the same as\nthe excitation probability pexc. In general, we can state that this probability\nshould depend on the amount of time the molecular ion spends in the tran-\nsition region: the longer is the time, the higher is the excitation probability.\nSince the time spent in the transition region is inversely proportional to the\nrelative velocity of H+ and He+ at r = Rx (in radial direction), the proba-\nbility pexc is a function of this relative (radial) velocity at position r = Rx:\n\npexc = pexc(vx), (128)\n\nwhere vx ≡ ṙ(r = Rx). The velocity vx is given by Eq. (67):\n\nvx =\ndr\n\ndt\n\n∣∣∣∣\nr=Rx\n\n= vr\n\n(\n1−\n\nV(Rx)\n\nECM\n−\nb2\n\nR2x\n\n)\n, (129)\n\nwhere V(Rx) can be either the ground state or the excited state potential,\ndepending on the state of the molecular ion when it enters the transition\nregion. In Eq. (129) vr is the relative velocity of the colliding particles before\nthe collision. At the energies where the electron capture plays an impor-\ntant role (above 1 keV for protons and above 100 eV for muons), the term\nV(Rx)/ECM can be neglected for both states, since the approximate values\nof the interaction potentials at r = Rx = 2.5 a.u. are V1(Rx) ≈ 0 eV and\nV2(Rx) ≈ 11 eV. Hence, Eq. (129) simplifies to:\n\nvx ≈ vr\n(\n1−\n\nb2\n\nR2x\n\n)\n. (130)\n\nAs the velocity vx depends on the initial relative velocity vr and on the\nimpact parameter b, also the excitation probability is a function of these two\nvariables5:\n\npexc(vx) = pexc(vr,b). (131)\n\nSince for each trajectory the transition region has to be passed twice (on\nthe way in and on the way out), the total probability Pexc(vr,b) of the exci-\ntation to the 2Σ state for each trajectory is given by\n\nPexc(vr,b) = pexc(vr,b) (1− pexc(vr,b)) , (132)\n\n5 The exact form of this function depends on the shape of the potential energy curves of the\nstates between which the transition is occurring. For several cases, analytical expressions\nfor this probability have been derived, for example, the so-called Landau-Zener-Stuckelberg,\nRosen-Zener and Demkov approximations [26, 34]. The latter two are applicable to the 1Σ−\n\n2Σ transition of the HeH+ system [34]. Both of these analytical expressions are consistent\nwith our argument that the excitation probability is a function of b and vr.\n\n\n\n42 theory of muon-helium collisions\n\ni. e., the electron capture takes place when there is an excitation during one\ncrossing of the transition region, but no de-excitation in the other crossing.\nThe total transition probability is the same for the two trajectories.\n\nHowever, there is one additional requirement for the transition to happen:\nthe projectile needs to actually pass through the transition region, i. e., the\nturning point r0 has to be smaller than the transition point Rx. At fixed\nenergy ECM, the position of the turning point is a function of the impact pa-\nrameter, as visible from Eqs. (125) and (127): the larger the impact parameter,\nthe larger is r0. This means that there is a critical value of the impact param-\neter above which the transition to the excited state is not possible. Since for\nlarge impact parameters r0 ≈ b, the critical value of the impact parameter\nis approximately given by bmax ≈ Rx. Thus, the total transition probability\nfor each trajectory becomes:\n\nPexc(vr,b) =\n\n{\npexc(vr,b) (1− pexc(vr,b)) for b < bmax\n\n0 for b > bmax\n. (133)\n\nFrom the deflection function χ(b) and the transition probability Pexc(vr,b),\nwe can obtain the differential cross section using Eq. (83). Because the exci-\ntation (electron capture) can occur in two different ways, as detailed before\n(see Fig. 11), for each impact parameter b there are two possible collision\noutcomes, leading to two different trajectories described by the deflection\nangles χ1(b) and χ2(b) (see Fig. 12). Conversely, we can turn this argument\naround, and state that a given angle of deflection χ can be produced by two\ndifferent impact parameters, b1 and b2, corresponding to the two different\nscattering trajectories. Classically, we can simply sum these two contribu-\ntions to obtain the angular differential cross section:\n\nIEC (χ) = Pexc(vr,b1)IEL,1 (χ) + Pexc(vr,b2)IEL,2 (χ) (134)\n\n= Pexc(vr,b1)\nb1\n\nsinχ\n\n∣∣∣∣db1dχ\n∣∣∣∣+ Pexc(vr,b2) b2sinχ\n\n∣∣∣∣db2dχ\n∣∣∣∣ . (135)\n\nThe total electron capture cross section σEC is obtained by integrating the\ndifferential cross section IEC over all deflection angles, which is equivalent to\nintegrating over all impact parameters b for which the transition is possible:\n\nσEC = 2π\n\n∫\nIEC (χ) sinχdχ (136)\n\n= 2π\n\n∞∫\n0\n\nPexc(vr,b1)b1 db1 + 2π\n\n∞∫\n0\n\nPexc(vr,b2)b2 db2 (137)\n\n= 4π\n\nbmax∫\n0\n\npexc(vr,b) (1− pexc(vr,b))bdb (138)\n\n≈ 4π\nRx∫\n0\n\npexc(vr,b) (1− pexc(vr,b))bdb. (139)\n\n\n\n2.4 low-energy regime 43\n\nThis finally leads us to the result that the total cross section for electron\ncapture depends only on the initial relative velocity of the two particles par-\nticipating in the collision:\n\nσEC = σEC(vr) = σEC(vA). (140)\n\nFrom Eq. (140) we can infer that the muon electron-capture cross section\nis the same as the proton electron-capture cross section when the muon\nvelocity is the same as the proton velocity (velocity scaling of the cross\nsections).\n\nTherefore, the stopping power due to electron capture SEC, determined\nfrom Eq. (96), also scales with the velocity, since the inelastic energy loss per\ncollision QEC depends only on the difference of ionization energies between\nHe (24.6 eV) and hydrogen (muonium) (13.6 eV):\n\nSEC(vA) = QEC · σEC(vA) = 11 eV · σEC(vA). (141)\n\nThe velocity-scaling of the cross section is a consequence of two important\nfeatures of the electron-capture collision:\n\n1. the transition probability is only a function of the initial relative veloc-\nity vr (and impact parameter),\n\n2. the critical value of the impact parameter above which no transition is\npossible does not depend on energy (it only depends on the character-\nistics of the interaction, i. e., on Rx).\n\nBoth of these statements rest on the requirement that V(Rx)/ECM � 1, i. e.,\nthat ECM � 10 eV . As the energy approaches 10 eV we can expect that\nthe proton and muon cross sections start to differ. Indeed, at lower energy,\nthe projectile mass significantly modifies the expressions for the velocity vx\nand the turning point r0, so that Pexc and bmax become mass-dependent as\nwell.\n\nBefore moving on to the low-energy elastic collisions, it is worthwhile to\nmention that the arguments used in this section for the electron-capture cross\nsection can be used also for other inelastic processes. In our case, the other\nimportant inelastic processes are the inelastic collisions of muonium (formed\nin the above described electron-capture process) with He atoms, such as Mu\nelectron loss and Mu excitation:\n\nMu+He→ µ+ +He+ e− (Mu electron loss) (142)\n\nMu+He→Mu∗ +He (Mu excitation). (143)\n\nThese processes involve a transition from the ground state of the HeMu\nmolecule to another (excited) state of the HeMu molecule, analogous to\nthe transition from ground to the first excited state of Heµ+ occurring in\nan electron-capture collision. The adiabatic potential energies of the HeH\nmolecule are shown in Fig. 13 (HeMu potential energy curves are not avail-\nable to our knowledge). Provided that the transition is possible only in a\nnarrow region around some r = Rx, as was the case for the electron-capture\n\n\n\n44 theory of muon-helium collisions\n\nFigure 13: Adiabatic potential energy curves for several low-lying states of the HeH\nmolecule. The x-axis is the radial separation between hydrogen and he-\nlium nuclei in atomic units. The higher excited states (Rydberg states)\napproach the potential energy curve of the ground state of the HeH+\n\nmolecular ion (continuum limit). Figure reproduced from [37].\n\nprocess, the transition probability would again be velocity-dependent and\nthere would also be a cut-off impact parameter above which the transition is\nnot possible. According to [38], the requirement for the transition region to\nbe narrow seems to be satisfied for the transitions between different states\nof the HeH molecule. Hence, we expect that the total inelastic cross sections\nfor Mu electron loss and Mu excitation depend only on the projectile (Mu)\nvelocity.\n\nlow-energy elastic collisions The elastic collisions of muons (pro-\ntons) with He atoms are governed by the ground state potential of the Heµ+\n\n(HeH+) molecular ion. The procedure to quantify the elastic scattering is the\nusual: we start by calculating the deflection function for the ground state\nmolecular potential 1Σ using Eq. (72), followed by determining the differen-\ntial cross section from Eq. (78). Even without calculating this cross section,\nwe know that it should only depend on the center-of-mass energy, since the\ndeflection function for elastic scattering is a function of the center-of-mass\nenergy only, as discussed in Section 2.2. Thus, the muon and proton differ-\nential cross sections (obtained from the deflection function) should be equal\n\n\n\n2.4 low-energy regime 45\n\nat the same center-of-mass energy. This is also true for the total and mo-\nmentum transfer cross sections, since they are obtained by integrating the\ndifferential cross section over all possible deflection angles. Since the colli-\nsions considered here are purely elastic, there is no cut-off angle χmin or\ntransition probability affecting this conclusion, as for the inelastic collisions.\n\nThe elastic stopping power can be determined from the momentum trans-\nfer cross section, according to Eq. (92):\n\nSEL(ECM) = 2\nmA\n\nmA +mB\nECM · σMT (ECM), (144)\n\nwhere formB here we have to take the He atom mass. This is the low-energy\nextension of the elastic (nuclear) stopping of the high-energy regime, which\nis given by Eq. (119). It follows the same scaling, given by Eq. (120). The\nmain difference between the low- and high-energy elastic collisions is in the\ncross section σMT . The momentum-transfer cross section is different for the\ntwo regimes since the elastic scattering is governed by a different interaction\npotential in each regime (screened Coulomb potential for the “high-energy”\nregime versus molecular ground state potential for the low-energy regime).\n\nThe distribution of the scattering angles is also different for low- and\n“high-energy” elastic collisions. At high energy (Ep > 10 keV , Eµ > 1 keV),\nthe elastic scattering is dominated by small-angle collisions, as described by\nthe “high-energy” differential cross section of Eq. (100). At low energy this\nis not necessarily the case. The projectile at low energy interacts with both\ntarget nucleus and target electrons at the same time, so that the interaction\npotential is not purely repulsive anymore, as shown in Fig. 14.\n\nThe shape of the ion-atom potential at low energy can be understood in\nthe following way: at small r, the interaction is dominated by the Coulomb\nrepulsion between the positive projectile (proton or muon) and the He nu-\ncleus. With increasing r the electrons distribute themselves partially in be-\ntween the ion and the He nucleus, creating an attractive interaction. The\npotential has a minimum at r = rm = 1.46 a.u. with a depth of V(rm) =\n\n−2.04 eV [39]. At this position, the repulsive and attractive forces are exactly\nbalanced.\n\nDue to the interplay between the attractive and the repulsive part of the\nground state potential curve, the deflection function has a more complicated\nbehavior compared to the deflection functions of monotonic potentials (see\nFig. 15). This is especially prominent at low energies. This behavior of the de-\nflection function has several consequences for the angular differential cross\nsection, one of them being that large-angle scattering becomes significant at\nlow energies, as we will detail below. Since the angular distribution will\nimpact strongly the muon drift through the muCool device, it is worthwhile\nto inspect the deflection function and the corresponding angular differential\ncross section in more detail.\n\nThe low-energy elastic deflection function and cross section As we can\nsee from Fig. 15, for small impact parameters, the deflection angle is positive,\nand it decreases with increasing impact parameter until it reaches zero. This\n\n\n\n46 theory of muon-helium collisions\n\nFigure 14: Effective potential Veff = V(r)+ECMb2/r2 for several values of ECMb2,\ni.e., for several centrifugal barriers. The potential V(r) is the ground state\n(1Σ) potential of the HeH+ molecular ion (see Fig. 10). The numerical\nvalues of the ground state potential were taken from [39]. Atomic units\nare used: for potential energy 1 a.u. = 27.2 eV, for radial separation\n1 a.u. = 0.529 Å.\n\nis consistent with Fig. 8 (right) as the scattering is dominated by the repulsive\ncore for small impact parameters. After that point, the deflection becomes\nnegative and it continues decreasing with increasing b until it reaches a\nminimum. The position and the depth of this minimum depend on the\ncenter-of-mass energy. After the minimum, the deflection function starts\nincreasing again and asymptotically (b→∞) it reaches zero.\n\nHence, the deflection function can be divided into three regions, as marked\nin Fig. 16. The region 1 is the region where the deflection function (angle)\nis positive. In region 2 and 3 the deflection angles are negative, with the\nregion 2 being on the left (smaller impact parameters), and the region 3 on\nthe right (larger impact parameters) side of the minimum. The impact pa-\nrameter b = bg marks the transition from region 1 to region 2, while the\nimpact parameter b = br marks the minimum of the deflection function and\nthus the transition between regions 2 and 3.\n\nTo aid the explanation of the deflection function shape, it is helpful to\ninspect the effective potential for few representative values of the impact pa-\nrameter from each region. The effective potential determines the location of\nthe turning point r0 (Veff(r0) = ECM), thus limiting the part of the potential\nthat contributes to the scattering.\n\n\n\n2.4 low-energy regime 47\n\nFigure 15: Sketch of the deflection functions χ(b) for elastic low-energy muon-He\n(proton-He) collisions for several center-of-mass energies (darker shades\nare lower energies). The deflection function can take both positive and\nnegative values. At low energy, large negative deflections angles occur\nalso for large impact parameters b. The qualitative form of the deflection\nfunction has been inspired by results in [27, 40].\n\nFigure 16: Sketch of a typical deflection function χ(b) for elastic low-energy muon-\nHe (proton-He) collisions. The three regions discussed in the main text\nare also indicated.\n\nThe effective potentials for several values of ECMb2, i.e., for several cen-\ntrifugal barriers are shown in Fig. 14. For large values of ECMb2, the ef-\nfective potential is purely repulsive as the attractive part of the potential is\ncompensated by the repulsive centrifugal barrier. As we lower the centrifu-\ngal barrier (smaller energy and/or impact parameter), the effective potential\n\n\n\n48 theory of muon-helium collisions\n\ndevelops an attractive well, that significantly impacts the collision dynamics\nat low energy.\n\nFigure 17: Trajectories in the center-of-mass frame during elastic low-energy muon-\nHe (proton-He) collisions, for several values of the impact parameters\nb and at ECM = 1 eV. The trajectories were calculated using Comsol\n“Charged particle tracing” module, starting from the ground state poten-\ntial of the HeH+ [39]. The trajectories for the three regions of the deflec-\ntion function defined in Fig. 16 are shown separately. The corresponding\ndeflection functions χ(b) are sketched on the left for clarity.\n\nLet us now consider the deflection function, region by region, starting\nfrom region 3:\n\n\n\n2.4 low-energy regime 49\n\nregion 3 : For large impact parameters, the scattering is dominated by the\ntail of the potential V(r), i.e., by the attractive part of the potential.\nThus, the deflection angles are negative, as shown in Fig. 17 (top),\nwith asymptotic behavior χ → 0 for b → ∞. The turning point for\nthese large impact parameters is located in the region where the poten-\ntial is attractive, so that two colliding particles do not approach close\nenough to feel the repulsive part of the potential. With decreasing im-\npact parameters, the turning point decreases and the colliding particles\nexperience a stronger attractive potential, resulting in larger negative\ndeflections. This behavior continues down to impact parameter b = br,\nwhere the maximum negative deflection χr is reached.\n\nregion 2 : As we decrease the impact parameter beyond b = br, the trajec-\ntories of the two colliding particles begin to be affected by the repulsive\npart of the potential. On the way to the turning point, the particles are\nfirst deflected by the attractive part of the potential (negative deflec-\ntion), but after they pass the potential minimum at r = rm they are\ndeflected in the opposite direction by the repulsive part of the poten-\ntial (positive deflection). So the initial deflection through the attractive\npart of the potential is partially canceled out by the repulsion at small r.\nThus, the absolute value of the deflection angle decreases with decreas-\ning impact parameter. Eventually, as we decrease the impact parame-\nter further down to b = bg, the deflections due to the attractive and\nrepulsive potential parts exactly cancel out, so that the total deflection\nis zero (χ(bg) = 0). Some selected trajectories (in the center-of-mass\nframe) from the region 2 are sketched in Fig. 17 (middle).\n\nregion 1 : By further decreasing the impact parameter (b < bg), the collid-\ning particles reach increasingly steeper part of the repulsive potential\ncore, so that the deflection angle is dominated by the repulsive part of\nthe potential, leading to increasingly larger positive deflection angles.\nSome trajectories typical for this region are shown in Fig. 17 (bottom).\n\nSuch a behavior of the deflection function gives rise to two singularities\nin the differential cross section. The first one, the so-called glory singularity\nat b = bg, is caused by the 1/ sinχ term in the Eq. (78) (as 1/ sinχ → ∞\nfor b → bg). The second one, the so-called rainbow scattering singularity,\nat b = br, is caused by the derivative dχ/db in Eq. (78), as dχ/db → 0 for\nb→ br.\n\nThe glory type of singularity causes the differential cross section to in-\ncrease for small scattering angles. However, such small deflections cause\nonly small energy losses, leaving the muon (proton) motion almost unaf-\nfected. More important for the muon (proton) motion is the contribution to\nthe differential cross section from the rainbow scattering, peaking at (larger)\nangles around χ = χr. As visible from Fig. 19, the differential cross section\nis also increased for angles 0 < χ < χr (compared to a purely repulsive\npotential), because for these angles several trajectories have the same final\ndeflection angle. An example of three impact parameters that produce the\nsame deflection angle χ is shown in Fig. 18. Also the relatively large impact\n\n\n\n50 theory of muon-helium collisions\n\nFigure 18: Sketch of a deflection function χ(b) for elastic low-energy muon-He\n(proton-He) collisions, showing that three impact parameters (b1, b2 and\nb3) can produce the same absolute value of the deflection angle (χ0). The\ninset shows three trajectories leading to the same deflection angle χ0. The\nfigure was inspired by [35]\n\nparameters (from regions 2 and 3) can cause a large deflection angle, thus\nincreasing further the differential cross section.\n\nDepending on the value of χr, relatively large scattering angles might\nbecome possible, implying large momentum transfer cross section and large\nelastic stopping power. The value of the rainbow angle χr depends strongly\non the center-of-mass energy [42]:\n\nχr ∝\nV(rm)\n\nECM\n. (145)\n\nThus, as we lower the collision energy, χr increases, so that large-angle scat-\ntering becomes likely. At a certain energy, χr → π, i. e., the particle makes a\nfull revolution around the potential center. For protons in He this happens\nat an energy of ECM ≈ 0.82 eV [40]. This small value is the consequence of\nthe shape of the effective potential for small centrifugal barriers. Indeed, for\nsmall centrifugal barriers (small ECMb2), the effective potential has a well,\nand a rather flat barrier. When the center-of-mass energy approaches the\nheight of this barrier, the relative radial velocity of the colliding particles in\nthis region becomes so small that the particles spend a considerable time\nat the same radial distance r (in the region where the interaction potential\n1Σ is attractive). In this time, the particles orbit the center-of-force, giving\nrise to large-angle scattering. With decreasing energy, the time spent by the\nparticles close to the potential barrier increases. Hence, the rainbow angle in-\ncreases with decreasing energy and eventually it becomes larger than π. As\na consequence, the scattering angle distribution (differential cross section)\nbecomes more and more isotropic.\n\n\n\n2.5 summary and validity of the classical scattering theory 51\n\nFigure 19: Sketch of a typical angular differential cross section for elastic low-energy\nmuon-He (proton-He) collisions at about ECM ≈ 1 eV. The peak (singu-\nlarity) in the cross section at the deflection angle χr is caused by the\nrainbow scattering. The position of the peak moves to larger deflection\nangles with decreasing center-of-mass energy. For angles smaller than\nthe rainbow angle χ < χr all three regions of the deflection function\ncontribute to the cross section, while for angles larger than χ > χr only\nregion 1 contributes. The qualitative form of the differential cross section\nwas inspired by the results of [27, 40, 41].\n\n2.5 summary and validity of the classical scattering theory\n\nUntil now, we have applied classical scattering theory to the various types of\ncollisions contributing to the muon energy loss. In the “high-energy” regime\n(Ep > 10 keV, Eµ > 1 keV) we have seen that the inelastic collisions dominate\nthe energy loss of the projectile. This inelastic (electronic) stopping power\nwas found to depend only on the projectile velocity, but not on the projectile\nmass. Thus, the stopping power is the same for muons and protons at the\nsame velocity (velocity-scaling). For low-energy (Ep < 10 keV, Eµ < 1 keV)\ninelastic collisions (electron-capture) we have obtained a similar result: the\ntotal inelastic cross section and the inelastic stopping power are the same for\nprotons and muons at the same velocity, provided the energy is high enough\n(Ep, Eµ > 100 eV).\n\nThe stopping power due to elastic collisions has different scaling from the\ninelastic stopping power: at the same center-of-mass energy the stopping\npower for protons in He is larger by a constant factor mp\n\nmp+mHe\n· mµ+mHe\n\nmµ\n≈\n\n7.3 than the stopping power for muons in helium. This is true for both\nlow-energy and high-energy elastic (nuclear) scattering (see Eq. (120)). This\nfactor is the consequence of the elastic collision kinematics: at the same\ncenter-of-mass energy and for the same center-of-mass deflection angle χ,\nthe proton will lose mp\n\nmp+mHe\n· mµ+mHe\n\nmµ\ntimes more energy than the muon, as\n\n\n\n52 theory of muon-helium collisions\n\nvisible from Eq. (45). Furthermore, we have demonstrated that differential,\ntotal and momentum transfer cross sections for the elastic collisions depend\nonly on the center-of-mass energy (energy-scaling), so they are the same for\nmuons and protons at the same center-of-mass energy.\n\nThe more exact treatment of the scattering would require quantum me-\nchanics. In a quantum-mechanical approach, the particle is represented by\na wave function. The analog of the classical deflection angle χ in quantum\nmechanics is the phase shift of the particle wave function [43]. This phase\nshift ηl is a function of the orbital quantum number l (angular momentum),\njust as the classical deflection function is a function of the impact parameter\n(χ(b)). The quantum-mechanical cross section is determined from this phase\nshift, similar to how the classical differential cross section is determined from\nthe deflection function. Some more details are given in Appendix C. In the\nsmall wavelength limit, the quantum-mechanical and the classical scattering\ncan be directly related via the so-called classical equivalence relation, which\nlinks the classical impact parameter b to the quantum orbital number l [43]:\n\nb =\nl+ 1\n\n2\n\nk\n, (146)\n\nand the classical deflection function χ(b) to the quantum phase shift ηl:\n\nχ(b) = 2\n∂ηl\n∂l\n\n=\n2\n\nk\n\n∂ηl\n∂b\n\n, (147)\n\nwhere k = mrvr/ h is the wave-number.\nWhen these relations are valid, the classical and quantum-mechanical scat-\n\ntering theory give the same results for the total and momentum cross sec-\ntions. However, the validity of the classical equivalence relation is limited\nby the Heisenberg uncertainty principle. In order for the concept of the im-\npact parameter and particle trajectory to be meaningful, the uncertainty in\nthe position of the projectile should be much smaller than the value of the\nimpact parameter, and the uncertainty in the angle should be much smaller\nthan the scattering angle χ. These two uncertainty requirements and there-\nfore, the classical equivalence relations of Eqs. (146) and (147) are valid if the\ndeflection angle is much larger than the critical angle χc [41]:\n\nχ�\n h\n\nmrvrb\n=\n\n h√\n2mrECM b\n\n≡ χc. (148)\n\nFor light particles, such as electrons, the semi-classical picture breaks down\nvery quickly, already at very large scattering angles. For heavier particles,\nsuch as protons and muons, the semi-classical picture is valid down to small\nangles, provided that the relative velocity vr is not too small. This limit is\nhowever not very stringent. Indeed it has been demonstrated [44] that for\npower-law potentials the classical differential cross section deviates from the\nquantum mechanical one by at most 2.5% only at the critical angle.\n\nUnderstanding the range of validity of the classical approach is important\nbecause the scaling relationships (between muons and protons) for cross\n\n\n\n2.5 summary and validity of the classical scattering theory 53\n\nsections and stopping power have been derived from a classical approach.\nBelow the critical angle χc the dependence of the differential cross sections\non the mass and velocity is usually not as simple as in the classical regime, so\nthat no exact scaling between muons and protons can be found. In principle,\nthen the full quantum mechanical calculation should be done specifically for\nmuons, instead of simply scaling proton data.\n\nIn the following, we will consider whether the quantum mechanical effects\ncould modify significantly our conclusion about the scaling of the stopping\npowers and cross sections for various collisional processes. Each energy\nregime will be treated separately.\n\n“high-energy” regime (Ep > 10 keV, Eµ > 1 keV)\nIn the “high-energy” regime the differential cross section describing the\n\nscattering is exactly the Rutherford cross section, assuming the projectile is\nscattering off a free electron or a nucleus. It is well known [28, 41] that\nthe Rutherford differential cross section obtained from a classical treatment\nof the collision is identical to the cross section obtained from a quantum-\nmechanical calculation. However, to obtain the correct electronic stopping\npower, the binding of the electrons needs to be taken into account. We have\ndone this in a very simplified way, by noting that the elastic energy transfer\nto the electron needs to be at least as large as the lowest possible excitation\nenergy of the target atom.\n\nThe more accurate (quantum-mechanical) treatment of the scattering off\nan electron bound in an atom yields the famous Bethe-Bloch formula [45]:\n\nSe = 4πZB ·\ne4\n\nmev\n2\nA\n\nL(vA), (149)\n\nwhere L(vA) is the so-called stopping number that replaces the logarithmic\nterm of Eq. (110). For non-relativistic particles, the stopping number is typi-\ncally written as [45]:\n\nL(vA) = L0(vA) +ZAL1(vA) +Z\n2\nAL2(vA). (150)\n\nThe first term L0 is similar to the logarithmic term of Eq. (110), however it\nincludes some additional corrections:\n\nL0(vA) = ln\n(\n2mev\n\n2\nA\n\nI\n\n)\n− ln\n\n(\n1−\n\nv2A\nc2\n\n)\n−\nv2A\nc2\n\n+∆Lshell(vA) +∆Ldensity(vA), (151)\n\nwhere I is the mean-excitation energy of the target atom, which requires av-\neraging over all possible excitations (and ionization) of the target atom. The\nmean-excitation energy is usually extracted from the experimental data. For\nhelium gas, I = 41.8 eV [46]. The shell correction ∆Lshell accounts for the or-\nbital velocity of the electrons and is of the order of few percent for protons in\nHe [47]. The term ∆Ldensity is the so-called density-effect correction, which\ntakes into account that, at high projectile energy, the projectile polarizes the\ntarget. For protons, this correction is about 1% around 500 MeV energy [48],\nand is even smaller at the energies we are concerned with.\n\n\n\n54 theory of muon-helium collisions\n\nThe other corrections in Eq. (150) are: the Barkas-Andersen term (L1),\nwhich accounts for the difference in stopping power of positively and neg-\natively charged projectiles (important only for proton energies between 0.6\nand 10 MeV [45]), and the Bloch correction (L2), which contributes about few\npercent at low energy [45].\n\nAll these corrections are again a function of the projectile velocity (and\ncharge), and thus should be the same for protons and muon of the same\nvelocity. Therefore, our conclusion that the stopping powers for muons can\nbe velocity-scaled from the proton stopping power in the high-energy regime\nis valid.\n\nelectron-capture We have seen that the electron capture is only pos-\nsible for relatively small impact parameters (approximately b < bmax =\n\n2.5 a.u.). Impact parameters larger than this critical value do not contribute\nto the total cross section, and thus, the deflection angles below some χmin(bmax)\ndo not enter into the total cross section (see also Fig. 12).\n\nIn order for the classical theory to be applicable, the angle χmin(bmax)\nneeds to be larger than the critical angle χc at b = bmax\n\n6, given by Eq. (148):\n\nχmin(bmax)� χc =\n h√\n\n2mrECM bmax\n(153)\n\nFrom [49], we find that the minimum laboratory scattering angle ϑA,min\n\ncontributing to the electron-capture cross section of protons in He at projec-\ntile energy of EA = 1.5 keV is ϑA,min = 0.12 deg. The corresponding center-\nof-mass values are χmin = 0.15 deg = 2.6 · 10−3 rad and ECM = 1.2 keV ,\nusing Eqs. (43) and (40). From Eq. (153) we find that at this energy the criti-\ncal angle is χc = 10−3 rad for protons and thus the requirement χmin � χc\nis satisfied.\n\nThis requirement can be checked for other values of ECM by noting that\nχc(ECM) ∝ 1/\n\n√\nECM and χmin(bmax) ∝ 1/ECM in the small-angle approxi-\n\nmation (justified here) of Eq. (74). Hence, the ratio of the critical to minimum\nangle scales with the center-of-mass energy (and also laboratory energy) as:\n\nχc\n\nχmin\n∝\n√\nECM ∝\n\n√\nEA. (154)\n\n6 If this is true for b = bmax, then it is automatically true for all impact parameters smaller\nthan bmax. This can be seen by comparing χc(b) at some fixed ECM to χ(b) at the same\nenergy. The small deflection angles for power-law potential (V(r) ∝ r−n) have the following\ndependence on the impact parameter: χ(b) ∝ b−n, as seen from the Eq. (75). Combined with\nEq. (153) we find that χc(b)\n\nχ(b) ∝ b\nn−1, so that:\n\nχc(b)\n\nχ(b)\n= (b/bmax)\n\nn−1 χc(bmax)\n\nχ(bmax)\n. (152)\n\nFor b < bmax it follows that χc(b)\nχ(b) <\n\nχc(bmax)\nχ(bmax)\n\nprovided that n > 1. This is true for the\ninteraction potentials of both the ground and the first excited state of the HeH+ molecular\nion, that show a r−4 dependence at large r. In general, this is true for realistic ion-atom\npotentials.\n\n\n\n2.5 summary and validity of the classical scattering theory 55\n\nUsing this scaling, at EA = 10 keV (which is roughly the upper limit of the\nlow-energy regime for protons) the ratio of angles is:\n\nχc\n\nχmin\n(10 keV) =\n\nχc\n\nχmin\n(1.5 keV) ·\n\n√\n10/1.5 = 0.38 ·\n\n√\n10/1.5 ≈ 1. (155)\n\nHence, at this energy we are at the limit of the validity of the classical picture.\nHowever, as mentioned before the deviation of the classical differential cross\nsection at χc from the quantum-mechanical cross section is about 2.5% at\nmost [44]. The difference between the classical and quantum mechanical\ntotal inelastic cross sections should be even smaller, because the integration\nof the differential cross section runs over all angles from χmin to π, and for\nχ > χmin the requirement χ � χc is even better fulfilled (see footnote 6).\nWe can conclude that the classical scattering theory can be reasonably well\napplied in the whole low-energy regime between about 100 eV and 10 keV\nproton energies.\n\nSimilar consideration can be applied also to muons. For muons, the mini-\nmum angle χmin contributing to the total inelastic cross section is the same\nas the χmin for protons at the same center-of-mass energy7. The upper\nlimit of the low-energy regime for muons is at EA = 1 keV . At this en-\nergy the minimum angle is χmin ≈ 3.1 · 10−3 rad, while the critical angle is\nχc ≈ 3 · 10−3 rad, so that χc/χmin ≈ 1.\n\nThus, our finding that the total electron-capture cross section and stopping\npower adhere to velocity scaling should hold both for protons and muons in\nthe low energy regime (1 keV < Ep < 10 keV, 100 eV < Eµ < 1 keV).\n\nlow-energy elastic collisions For low-energy elastic collisions, the\nfull range of scattering angles contributes to the total elastic cross section, in\ncontrast to the inelastic collisions. The most significant contribution to the\ntotal cross section indeed comes from very small angles, except in the eV\nenergy range, where also large-angle scattering becomes likely.\n\nTherefore, the classical total elastic cross section differs significantly from\nthe quantum mechanical cross section. In this case, it is not possible anymore\nto relate muon and proton total elastic cross sections - no scaling necessarily\nexists between the two.\n\nHowever, we have seen before that the slowing-down of a charged particle\nvia elastic scattering can effectively be described with the momentum trans-\nfer cross sections, as visible from Eq. (92). The momentum-transfer cross\nsection is obtained by integrating the differential angular cross section over\nall angles, but weighted by the factor 1 − cosχ, i. e., only large scattering\nangles contribute significantly to the momentum-transfer cross section8.\n\nSince only large angles contribute to the momentum transfer cross section,\nthe use of the classical scattering theory is adequate also for the low-energy\nelastic scattering. Thus, we can use energy-scaling, as demonstrated before\n\n7 The deflection function depends on the center-of-mass energy only, as seen in Section 2.2.\n8 At low energy, the muon (proton) undergoes many collisions with the He atoms, however,\n\nonly collisions with significant angular scattering (and therefore significant energy loss) affect\nthe muon trajectory significantly, justifying the use of momentum transfer cross sections to\ndescribe the muon elastic collisions with He.\n\n\n\n56 theory of muon-helium collisions\n\nusing the classical approach, to adapt the momentum cross sections for pro-\ntons to muon collisions.\n\n\n\n3\nG E A N T 4 S I M U L AT I O N S O F T H E M U O N S T O P P I N G A N D\nD R I F T I N H E L I U M G A S\n\nThis chapter describes how the findings of Chapter 2 can be applied to the\nsimulation of the muon slowing-down and drift in He gas. First, the pro-\ncesses that are implemented for muons in the standard packages of Geant4\nare summarized. The limitation of the standard Geant4 muon processes is\ndemonstrated by comparing the stopping power computed with the stan-\ndard Geant 4 simulation with the stopping power obtained from SRIM [47].\n\nTo overcome these limitations, low-energy (Eµ < 1 keV) muon-He colli-\nsions, as described in Chapter 2, have been implemented in Geant4 starting\nfrom proton data and appropriately scaled to muons. After a short overview\nof the available proton data, this chapter presents the implementation of the\nvarious muon-He interactions at energies Eµ < 1 keV in Geant4. To better\ngrasp, and also to partially validate the muon motion in He gas, several\nsimple scenarios were simulated and presented in the second part of this\nchapter:\n\n1. Muons stopping in helium gas, with neither electric nor magnetic\nfields applied. The ranges for various initial muon energies obtained\nfrom the Geant4 simulations are then compared with tabulated SRIM\nranges.\n\n2. Muon drift in He gas with uniform electric field applied. The results\nare compared to the theory developed in [50]. This scenario is the basis\nof the longitudinal stage of the muCool device.\n\n3. Muon drift in crossed electric and magnetic fields for various gas den-\nsities and electric field strengths. The muon drift in such fields is the\nbasis of the transverse compression stage.\n\n3.1 implementation of low-energy muon-he interactions in\n\ngeant4\n\nThis section explains how the findings of Chapter 2 pertaining to the muon-\nHe cross sections and energy losses in the various collisional processes can\nbe applied to the simulation of the muon slowing-down and drift in He. The\nsimulation is done using Geant4 [51], which is a collection of C++ libraries,\ncommonly used for Monte-Carlo simulation of the particle interactions with\nmatter in nuclear and high-energy particle physics. These libraries allow\ntracking of the particles in different materials, and also in the electric and\nmagnetic fields. It is possible to implement complex geometries, thus allow-\ning the simulation of the realistic experimental setup (including detectors).\nA large number of particle-medium interactions are already implemented in\n\n57\n\n\n\n58 geant4 simulations of the muon stopping and drift in helium gas\n\nGeant4. However, these are often aimed only at high-energy (MeV, GeV re-\ngion) applications and are thus not necessarily adequate at the low energies\nwe are concerned with.\n\nFor muons, Geant4 includes following interactions (“physics processes”):\n\n• “G4MuIonisation”, which is responsible for the muon energy loss due\nto the interactions with the target electrons (ionization and excitation).\nThis is implemented as continuous energy loss along each simulation\nstep, without any deflection.\n\n• “G4MuMultipleScattering” incorporates the changes of the muon direc-\ntion due to the elastic collisions with the target atoms at the end of\neach simulation step, without any (elastic) energy loss. Several models\nare available for calculating the net deflection of the muon during the\nstep (which might include multiple collisions depending on the step\nsize) in an efficient way.\n\n• “G4MuBremsstrahlung” adds the muon energy loss due to Bremsstrahl-\nung. This process is only relevant at much higher energies than the\nones we are interested in.\n\n• “G4MuPairProduction” includes the muon energy loss due to produc-\ntion of electron-positron pair. This process is also only relevant at\nenergies exceeding our range of interest.\n\nBeside these processes, the particle motion in electric and magnetic fields\nand the muon decay are included in Geant4 by default.\n\nTo check whether the above listed processes are sufficient to simulate the\nmuon drift in helium gas down to eV energies, we compare the total stop-\nping power produced by these processes to the stopping power from SRIM.\nThe SRIM electronic stopping power is velocity-scaled from proton data,\nwhile the nuclear stopping is scaled from protons according to Eq. (120).\nThis comparison is shown in Fig. 20.\n\nWe can see that at high energies (& 1 keV) the agreement between SRIM\nand Geant4 is sufficiently good. In fact, below 200 keV, Geant4 uses em-\npirical stopping power from the ICRU49 report [48], which is the same as\nthe SRIM data. Above 200 keV Geant4 uses theoretical Bethe-Bloch for-\nmula, Eq. (149), but with a small additional correction, needed to have a\nsmooth transition between empirical and theoretical stopping power curve\nat 200 keV. This additional correction is the reason for the small discrepancy\nbetween SRIM and Geant4 above about 100 keV. Yet, this small discrepancy\ndoes not affect the simulation of the muon motion in our device significantly,\nsince muons spend only a short amount of their total drift time at these en-\nergies.\n\nHowever, below 1 keV the stopping powers of Geant4 and SRIM are sig-\nnificantly different. The reason is that Geant4 assumes the stopping power\nto be linearly proportional to the muon velocity down to eV energies. As\ndiscussed before, for muons in helium this model is not correct for energies\nbelow about 1 keV due to the threshold effect (at low energy there is not\n\n\n\n3.1 implementation of low-energy muon-he interactions in geant4 59\n\nFigure 20: Stopping power for muon in He as a function of the muon energy in\nthe laboratory frame, computed using standard Geant4 processes (black\ncontinuous line), compared with electronic and nuclear stopping pow-\ners from SRIM [47] (dashed lines). The Geant4 simulation includes fol-\nlowing physics processes: “G4MuIonisation”, “G4MuMultipleScattering”,\n\n“G4MuBremsstrahlung” and “G4MuPairProduction”. The SRIM electronic\nstopping power for muons is the velocity-scaled (Eq. (111)) SRIM elec-\ntronic stopping power for protons in He, while the muon nuclear stop-\nping power is obtained by scaling the SRIM nuclear stopping power for\nprotons in He using Eq. (120).\n\nsufficient energy to excite or ionize the target atoms). This deviation from\na linear velocity dependence was confirmed both experimentally and theo-\nretically for protons in helium [29, 31–33]. Moreover, in Geant4, the elastic\ncollisions with helium at low energy are completely absent. To simulate\nthe muCool device it is of utmost importance to include these collisions as\nthey govern the phase space compression of the muon beam, as explained\nin Chapter 1. At these low-energies the standard Geant4 processes are thus\nnot sufficient for our purposes.\n\nTherefore, we have to implement the low-energy processes (Eµ < 1 keV)\nin Geant4 ourselves. For this purpose, we need the cross section data for the\nvarious low-energy muon-helium interactions. Since the required data does\nnot exist for muons we have to obtain the muon cross sections by scaling\nproton data according to the findings of the previous chapter. In the follow-\ning we will examine the low-energy cross sections available for protons in\nhelium and how we can adapt them for muons.\n\nProton cross section data The proton-helium and hydrogen-helium cross\nsections which were used in this thesis (after appropriate scaling) to simulate\n\n\n\n60 geant4 simulations of the muon stopping and drift in helium gas\n\nthe muon drift for energies Eµ < 1 keV are plotted in Fig. 21 as a function\nof the center-of-mass energy. The following cross sections were included:\n\n• momentum-transfer cross section σMT for proton-helium elastic col-\nlisions in the energy range from 0.1 eV to 10 keV from Ref. [52], calcu-\nlated starting from the Born-Oppenheimer HeH+ ground state poten-\ntial, like the one of Fig. 10.\n\n• electron-capture cross section σcapt for protons in helium, for en-\nergies ranging from about 10 eV to 10 MeV [53]. This cross section\nincludes the capture into ground, as well as into the all other excited\nstates of hydrogen. It was obtained [53] by fitting an energy-dependent\nanalytic formula to the available experimental data. The relative root-\nmean-square deviation of the data from the fitted cross section is 54%,\nmainly due to the scatter of the experimental data and use of a simpli-\nfied fitting function.\n\n• electron-loss cross section σloss for hydrogen in helium, for energies\nfrom about 10 eV to 10 MeV, obtained from the same compilation [53]\nas the proton electron-capture cross section. The relative root-mean-\nsquare deviation of the fitted electron-loss cross section from the ex-\nperimental data is 14%.\n\nAs concluded in the previous section, the momentum-transfer cross sec-\ntion can be energy-scaled to obtain the corresponding muon cross section,\nwhile all other (inelastic) cross sections are to be velocity-scaled. The scaled1\n\ncross sections are shown in Fig. 22. We can see that the charge exchange\n(electron-capture + electron-loss) is the dominant process between several\nhundred eV and 10 keV. At energies below 100 eV , all the inelastic processes\nare highly suppressed and thus only the low-energy elastic collisions play\nan important role. Using these cross sections, we can calculate the elastic\nand the inelastic stopping powers and compare them with the SRIM data of\nFig. 20.\n\nThe elastic stopping power SEL can be calculated using Eq. (92) and the\nmomentum-transfer cross sections of Fig. 22. The resulting elastic stop-\nping power is shown in Fig. 23 (solid blue line). The corresponding elas-\ntic (nuclear) stopping power from SRIM is indicated with a dashed blue\nline. The two stopping powers are of the same order of the magnitude, but\ntheir shapes differ. The difference comes from the fact that, to obtain the\nmomentum-transfer cross section, SRIM uses the so-called universal inter-\natomic potential [45], instead of the precise Born-Oppenheimer potential for\n\n1 Note that we used the velocity-scaling for the electron-loss, electron-capture and He ioniza-\ntion in the full energy range, even though this kind of scaling most likely is not valid as the\nenergies approach 10 eV. However, as we can see in Fig. 22, below 100 eV the elastic scatter-\ning becomes the most dominant process, and thus the simulations of the slowing-down are\ninsensitive to the exact value of the inelastic cross sections at these energies (yet it could be\nrelevant for the correct estimation of the muon losses). Moreover, as mentioned already, the\nempirical cross sections have large uncertainty (especially at low energies) given by some sys-\ntematical inconsistency between various measurements, so that the uncertainty of the scaling\nat these low energies is negligible.\n\n\n\n3.1 implementation of low-energy muon-he interactions in geant4 61\n\nFigure 21: Proton (hydrogen) cross sections in He versus center-of-mass energy. The\nproton electron-capture and hydrogen electron-loss cross sections were\ntaken from Ref. [53], while the moment-transfer proton-He cross section\nwas taken from Ref. [52]. These cross sections were used in the Geant4\nsimulations, after appropriate scaling to muons (see text and Fig. 22).\n\nFigure 22: Muon (muonium) cross sections in He versus center-of-mass energy,\nscaled from the proton (hydrogen) cross sections of Fig. 21. The muon\nelectron-capture and muonium electron-loss cross sections were velocity\nscaled from the corresponding proton and hydrogen cross sections, while\nthe muon-He momentum-transfer cross section was taken to be the same\nas the proton-He momentum-transfer cross section at the same center-of-\nmass energy.\n\n\n\n62 geant4 simulations of the muon stopping and drift in helium gas\n\nHeH+. The SRIM potential is essentially a screened Coulomb potential of\nthe form V(r) = qAqB\n\nr Φ(r), whereΦ(r) is the screening function. Such poten-\ntial allows for quick calculations of the nuclear stopping powers for various\nprojectile-target combinations, but at low energies it does not describe the\ninteraction correctly, since it is purely repulsive.\n\nFigure 23: Elastic stopping power (solid blue line) for muon in He as a function of\nthe muon energy in the laboratory frame, computed using Eq. (92) and\nmomentum-transfer cross section of Fig. 22. The inelastic stopping power\ndue to the charge-exchange process (red continuous line) is shown as\nwell and was obtained using Eq. (165) in combination with the electron-\ncapture and electron-loss cross sections of Fig. 22. The electronic and\nnuclear stopping powers from SRIM [47] (dashed lines) are obtained in\nthe same way as for Fig. 20.\n\nWe have seen in Sec. 2.4 that the attractive component of the Born-Oppen-\nheimer potential for the HeH+ ground state increases significantly the large-\nangle elastic scattering (because low-energy muon can orbit the He atom for\nsome time), resulting in the enhancement of the momentum-transfer cross\nsections. The structures in the differential angular cross sections due to\nsuch orbiting have been confirmed experimentally for protons in helium at\nenergies below 1 eV [54]. As these measurements fit well with the theoret-\nical calculations based on the Born-Oppenheimer ground state potential of\nHeH+, we can conclude that the cross sections of [52] are better suited for\nthe modeling of the low-energy elastic collisions than the purely repulsive\npotential assumed in SRIM. Indeed, the purely repulsive repulsive potentials\ncan not produce any orbiting effects.\n\nLet us now consider the stopping power due to the inelastic collisions,\nwhich are the dominant energy loss mechanism at muon energies above\n100 eV . At these energies the muon goes through a series of charge-exchange\n\n\n\n3.1 implementation of low-energy muon-he interactions in geant4 63\n\ncycles, thus spending a significant part of time as a neutral particle (muo-\nnium). The fraction of time it spends as muonium depends on the cross\nsection for electron-loss and electron-capture at a certain energy [55]:\n\nfMu(ECM) =\nσcapt(ECM)\n\nσloss(ECM) + σcapt(ECM)\n. (156)\n\nSimilarly, the fraction of time spent as positively charged muon is given\nby [55]:\n\nfµ+(ECM) =\nσloss(ECM)\n\nσloss(ECM) + σcapt(ECM)\n. (157)\n\nWhen the electron-capture cross section σcapt is larger than the electron-\nloss cross section σloss, the muonium fraction will be large, and vice-versa.\nOne important feature visible in Fig. 22 is that, at low energies (< 1 keV),\nthe electron-loss cross section σloss is much larger than the electron-capture\ncross section σcapt. Hence, any muonium formed at this low energy is quite\nlikely to be re-ionized, so that muon losses due to the muonium formation\nare small. In summary, high-energy muons slowing down and stopping\nin helium gas spend a significant fraction of time in the muonium state at\nkeV energies, but when slowing-down further to the eV regime, basically all\nformed muonium is re-ionized, so that it emerges again as free µ+. This was\nexperimentally verified using µSR techniques [56].\n\nEach time the charge-exchange collision happens (either electron-capture\nor electron-loss) the muon loses part of its kinetic energy. For electron-\ncapture process, the stopping power is given by Eq. (96):\n\nScapt(ECM) ≈ Qcaptσcapt(ECM), (158)\n\nwhere Qcapt is the average inelastic energy loss in the electron-capture col-\nlision. According to [57, 58], for energies Eµ < 1 keV (Ep < 10 keV), the cap-\nture to the hydrogen (muonium) in the ground state is much more probable\nthan a capture in one of the excited states. Thus, we can take the inelastic\nenergy loss for electron-capture to be approximately equal to the energy loss\nassociated with the capture into the ground state: Qcapt ≈ 11 eV , so that\nthe stopping power becomes:\n\nScapt(ECM) ≈ 11 eV · σcapt(ECM). (159)\n\nFor the electron-loss process, the expression is similar:\n\nSloss(ECM) ≈ Qlossσloss(ECM). (160)\n\nWe approximate the average inelastic energy loss with the ionization en-\nergy of muonium2: Qloss ≈ 13.6 eV . Thus, the stopping power becomes:\n\nSloss(ECM) ≈ 13.6 eV · σloss(ECM). (161)\n\n2 The actual inelastic loss is somewhat larger then the assumed ground-state binding energy of\n13.6 eV, as the ejected electron carries away some kinetic energy. To implement correctly this\nadditional energy loss into Geant4, we would need to have the differential cross section for\nthe ejected electron kinetic energy distribution, which are not available, to our knowledge.\nThere are some indications that the electrons are ejected with velocity close to projectile\nvelocity [55]. This contribution has not been included in the Geant4 simulation.\n\n\n\n64 geant4 simulations of the muon stopping and drift in helium gas\n\nTo evaluate the total stopping power produced by the electron-loss and\nelectron-capture collisions, we need to weigh each contribution by the frac-\ntion of time the muon spends in each charge state at a given energy, given\nby Eqs. (156) and (157):\n\nSe(ECM) = fµ+(ECM)Scapt(ECM) + fMu(ECM)Sloss(ECM) (162)\n\n=\nσloss(ECM)\n\nσloss(ECM) + σcapt(ECM)\nScapt(ECM) (163)\n\n+\nσcapt(ECM)\n\nσloss(ECM) + σcapt(ECM)\nSloss(ECM) (164)\n\n= 24.6 eV ·\nσloss(ECM)σcapt(ECM)\n\nσloss(ECM) + σcapt(ECM)\n. (165)\n\nThe stopping power due to the electron-capture and electron-loss pro-\ncesses, calculated using Eq. (165), is plotted in Fig. 23 (red solid line), along\nwith the electronic stopping power from SRIM (red dashed line). Obviously,\nalso in the range where the charge exchange is dominant (below 1 keV), the\nelectron-capture and electron-loss alone do not account fully for the total\nenergy loss of the muons in He, demonstrated by the fact that the charge-\nexchange stopping power is lower than total stopping power calculated with\nSRIM.\n\nIn principle, all other inelastic collisions of muons and muonium should\nalso be included to correctly reproduce the total stopping power. Most likely,\nthe main missing contribution to the stopping power is the muonium excita-\ntion process (Mu+He → Mu∗ +He), since in this energy range it has sim-\nilar cross section as muonium electron-loss (Mu+He → µ+ + e− +He), as\ncan be seen by comparing the corresponding cross sections for protons [59].\nThe cross sections for helium excitation (µ+ +He→ µ+ +He∗, Mu+He→\nMu+He∗) and ionization (µ+ +He → µ+ +He+ + e−, Mu+He → Mu+\n\nHe+ + e−) by both muon and muonium should be about 1-2 orders of mag-\nnitude smaller than the already described inelastic cross sections in this en-\nergy range (100 eV-1 keV) [57, 60, 61]. Therefore, they should not contribute\nsignificantly to the stopping power at these energies [62, 63].\n\nIn the simulations carried out within this thesis, instead of implement-\ning these processes directly, we chose a simpler approach: an additional\ninelastic energy loss has been added to the electron-capture and electron-\nloss processes to match approximately the SRIM inelastic stopping power.\nThis additional energy loss ranged from about 12 eV (on average) at 1 keV\nmuon energy down to 0 eV at 300 eV muon energy. In this way, the stop-\nping power from SRIM could be reproduced sufficiently well, as we will\nshow in the next subsection. Such simplification is acceptable in this energy\nrange, since muons only spend a small amount of time at these energies\n(basically only during the slowing-down process). The minor deviation in\nthe implemented stopping power would only alter slightly the initial muon\nstopping distribution in the He gas, without affecting the subsequent muon\ndrift, which is governed by the low-energy elastic collisions.\n\nBelow 300 eV, no additional energy loss is included in the charge ex-\nchange processes, as no stopping power data are available for the muons\n\n\n\n3.1 implementation of low-energy muon-he interactions in geant4 65\n\n(protons). Hence, at this energy, it is not clear how much should these in-\nelastic processes, besides charge exchange, contribute to the slowing-down\nof the muon (proton).\n\nFinally, it should be noted that, for the correct estimation of the muon\ncompression efficiency from the Geant4 simulation, it is important to also in-\ncorporate muonium elastic scattering (Mu+He→Mu+He). Even though\nthis process does not contribute to the total stopping power significantly, it\ncould affect the number of muons that might be permanently lost due to\nthe muonium formation. Indeed, if muonium with energy just above the\nelectron-loss threshold of 13.6 eV scatters elastically before being re-ionized,\nits energy can drop below this threshold where no electron-loss process is\npossible. The muon thus remains bound in the muonium, that simply ther-\nmalizes to the He thermal energy until it decays.\n\nSince no theoretical predictions exist for muonium-He (hydrogen-He) mo-\nmentum-transfer cross sections, in the Geant4 simulation we have assumed\nthat these are equal to the muon-He (proton-He) momentum-transfer cross\nsections.\n\nAfter having discussed the available proton cross sections and how those\nhave been adapted to muons, we will move on to describe the concrete im-\nplementation of these low-energy cross sections in Geant4.\n\nImplementation of the low-energy muon cross sections in Geant4 In\nthe simulation of the muon slowing-down and drift at energies higher than\n1 keV, we use the standard Geant4 processes for muons that were described\nin the beginning of this chapter. Below 1 keV we turn off the standard Geant4 The standard\n\nGeant4 processes\ncan be turned off at\ncertain energy by\nslightly modifying\nthe source code of\neach process. The\nGeant4 function\nSetActivationLow-\nEnergyLimit() can\nbe used for this\npurpose.\n\nprocesses and implement the low-energy elastic collisions and charge-ex-\nchange processes using the cross sections of Fig. 22. The implementation\nwhich will be explained in the following is based on the code of [19].\n\nIn Geant4, the Monte-Carlo simulation is done in steps. At each simu-\nlation step, only one of the various “competing” low-energy processes can\nhappen. Which process is selected depends on the cross sections of the vari-\nous processes.\n\nHere, the low-energy muon (muonium) interactions are implemented as\ndiscrete Geant4 processes, i. e., they are simulated collision by collision. Be-\ntween two consecutive collisions the muon (muonium) will propagate “free-\nly”. The energy loss and the direction change due to the collision occur only\nat the end of the step, i. e., at the time and position of the collision. The av-\nerage distance λi the muon travels between two collisions of a certain type i\n(i = elastic collision, electron-capture or electron-loss), called the mean free\npath, is inversely proportional to the cross section σi of the process i:\n\nλi(Eµ) =\n1\n\nNσi(Eµ)\n, (166)\n\nwhere N is the gas density (He atoms per cm3). The probability that the\nmuon undergoes a collision of type i after it has traveled a distance `i is\ngiven by the following expression:\n\nPi(`i) = 1− e\n−\n`i\nλi , (167)\n\n\n\n66 geant4 simulations of the muon stopping and drift in helium gas\n\ni.e., the distance `i between two collisions of type i has an exponential dis-\ntribution with mean λi. Since in Geant4 several processes are “competing”\nwith each other, several steps (collisions of different type) can occur between\ntwo collisions of the same type i. Therefore, the muon energy between two\ncollisions of the same type i can be changed by collisions of different type\n( 6= i). Hence, because of its energy dependence, the mean free path λi of the\nprocess i will also change3. However, the number of mean free paths nλ,i\n\nthat the muon travels between two collisions of the same type i is indepen-\ndent of the energy (and material) [64]. The number of mean free paths is\ndefined as:\n\nnλ,i =\n∑\nj\n\n∆`j\n\nλi,j\n, (168)\n\nwhere ∆`j is the length of j-th step, and λi,j is the mean free path of the\nprocess i at the beginning of the step j. The sum runs over all steps j (col-\nlisions) between two collisions of the type i. Generalizing Eq. (167) for the\ncase where several process are present, we can express the probability that\nthe muons undergoes a collision i after traveling nλ,i mean free paths as:\n\nPi = 1− e\n−nλ,i , (169)\n\ni. e., the number of mean free paths a muon travels before a certain collision\ntype occurs has an exponential distribution with mean value of 1 (on average\na muon will travel one mean free path between two collisions of the same\ntype).\n\nThe competition of the various processes (collision types) is handled using\nthe concepts introduced above. Initially, for each muon and each possible\nprocess, a number nλ,i is randomly generated according to the exponential\ndistribution with mean value of 1. Each process i proposes then the follow-\ning step length for the next simulation step:\n\n∆`i = nλ,i λi. (170)\n\nThe process with the shortest step length ∆`i will occur as the next step,\nincurring energy loss and direction change of the muon.\n\nThe next simulation step is handled in the following way: if the collision\nthat occurred in the previous step is of the type m (i = m), the number of\nmean free paths nλ,m will be generated anew from the exponential distribu-\ntion. For all other processes i 6= m (that have not occurred in the previous\nstep), the mean free path is updated in the following way:\n\nnλ,i → nλ,i −\n∆`m\n\nλi\n, (171)\n\nwhere ∆`m is the length of the previous step, defined by the process m, i. e.,\nthe fraction of the mean free path traveled during the step is subtracted from\nnλ,i. The proposed step length for the next step can be then computed anew\n\n3 The mean free path also changes if muon travels through different materials, or different gas\ndensities in-between two collisions.\n\n\n\n3.1 implementation of low-energy muon-he interactions in geant4 67\n\nusing Eq. (170), according to the updated (new) number of mean free paths.\nAgain, the process with the shortest proposed step length will occur at the\nnext simulation step and the procedure described above will be repeated\nuntil the end of the simulation4.\n\nIf the low-energy elastic collision is the process with the smallest step size,\nboth the muon (muonium) direction and kinetic energy are changed at the\nend of the step. To determine the elastic energy loss and the new direction,\nfirst the deflection angle χ in the center-of-mass frame is randomly generated\naccording to the cosχ distribution (i.e., the cosχ is uniformly distributed\nbetween −1 and 1). The elastic energy loss T is then calculated using Eq. (44):\n\nT = 2\nmµmHe\n\n(mµ +mHe)\n2\nEµ (1− cosχ) , (172)\n\nwhere Eµ is muon kinetic energy in the laboratory frame. After updating\nthe muon (muonium) kinetic energy ( Eµ → Eµ − T ), the muon momentum\nvector is rotated to the new direction. To determine the new direction, the\ncenter-of-mass scattering angle χ is first transformed to the scattering angle\nθ in the laboratory frame using Eq. (43):\n\ntan ϑ =\nsinχ\n\ncosχ+ mµ\n\nmHe\n\n, (173)\n\nand an additional (azimuthal) angle ϕ is generated uniformly on the interval\nbetween 0 and 2π. The muon momentum direction is then rotated\n\nfrom p = p\n\nexey\nez\n\n to p ′ = p ′\n\ne\n′\nx\n\ne ′y\n\ne ′z\n\n , (174)\n\nusing the following transformation [65]:\n\ne ′z = ez cos ϑ+ sin ϑ cosϕ ·\n√\n1− e2z (175)\n\ne ′y =\n1\n\n1− e2z\n\n[\ney(cos ϑ− eze ′z) + sin ϑ sinϕ · ex\n\n√\n1− e2z\n\n]\n(176)\n\ne ′x =\n1\n\n1− e2z\n\n[\nex(cos ϑ− eze ′z) + sin ϑ sinϕ · ey\n\n√\n1− e2z\n\n]\n(177)\n\nwhere ex, ey and ez are the direction cosines before the collision, while e ′x,\ne ′y and e ′z are the direction cosines after the collision.\n\nIf, however, the process with the smallest step was electron-capture or\nelectron-loss, the muon (muonium) charge and the kinetic energy are modi-\nfied, while its direction remains unaffected by the collision. For an electron-\ncapture collision the charge after the collision is 0, while after an electron-\n\n4 The simulation of one event (one muon) can end in several ways: after a certain tracking\ntime (commonly 10 µs), when the muon decays, or when it reaches the wall of our muCool\ndevice.\n\n\n\n68 geant4 simulations of the muon stopping and drift in helium gas\n\nloss collision the charge is 1. The kinetic energy is decreased by an amount\nQ, with:\n\nQloss = 13.6 eV +∆Q (178)\n\nQcapt = 11 eV +∆Q, (179)\n\nwhere the additional inelastic energy loss ∆Q, given by:\n\n∆Q =\n\n{\n220 MeV cm2 g−1 · ρ · ` · Eµ−300 eV700 eV for Eµ > 300 eV\n\n0 for Eµ < 300 eV\n(180)\n\nis added at muon energies above 300 eV in order to reproduce the SRIM\nstopping power, as explained in the previous section. Here, ρ is the helium\ngas density (in g/cm3) and ` is the distance the muon has traveled since the\nlast charge-exchange collision of the same type.\n\n3.2 muon range and slowing-down in he gas\n\nIn this section, the slowing-down of muons in helium gas (with no electric\nor magnetic field applied) will be simulated in Geant4 using the procedure\ndescribed in the previous section. The simulations will be compared with\nthe results from SRIM.\n\nThe stopping of a muon beam in 10 mbar helium gas at 10 K (density of\n4.8 · 10−5 g/cm3) was simulated for various initial muon energies, ranging\nfrom 2 keV to 4 MeV. For each initial energy, 104 muons were generated at\nz = 0 with momentum pointing in the +z-direction.\n\nThe evolution of the muon kinetic energy with time is shown in Fig. 24, for\nmuons starting with 4 MeV energy at t = 0. The color histogram shows the\nmuon energy versus time at each simulation step, while the black (central)\nline is the average time it takes for the muons to slow down to a certain\nenergy. On average, the slowing-down from 4 MeV down to 0.1 eV takes\naround 500 ns. The time spent slowing down to 1 keV is 480 ns, while\nslowing down from 1 keV to 0.1 eV takes only 25 ns.\n\nFrom this simulation, we can reconstruct the total stopping power the\nmuons experience during the slowing-down by dividing the average energyThe averaging is\n\ndone over all\nsimulation steps at\n\na certain energy.\n\nloss with the average step length at a certain energy. The total stopping\npower obtained in this way is shown in Fig. 25. For comparison, the SRIM\nnuclear and electronic stopping power are also shown in the same plot. We\ncan see that the stopping power, based on our custom low-energy exten-\nsion of the Geant4 simulation, matches the SRIM data considerably better\nthan the stopping power obtained using only the standard Geant4 processes,\nshown in Fig. 20.\n\nThere are two minor deviations from the SRIM electronic stopping power.\nThe first one is at around 1 keV muon energy, where the simulation switches\nfrom the standard Geant4 processes to the custom low-energy processes.\nThis small discrepancy could be fixed by making the switch to low-energy\nprocesses already at slightly higher energy (≈ 2 keV). The second deviation\nappears between 300 and 500 eV. It arises from the approximate way the\n\n\n\n3.2 muon range and slowing-down in he gas 69\n\nFigure 24: The evolution of muon energy versus time for 104 muons of 4MeV initial\nenergy. The color histogram shows the energy of the muons versus time\nat each simulation step, while the black (central) line is the average time\nit takes for the muons to slow down to a certain energy\n\nadditional energy loss was implemented in Geant4 (see Eq. (180)). However,\nthese deviations do not affect the accuracy of the simulation of the muon\nslowing-down process significantly, as we will show in the following.\n\nWe can test whether the stopping power resulting from the Geant4 sim-\nulation matches the SRIM stopping power sufficiently well by comparing\nthe projected muon ranges from the Geant4 simulation and from SRIM. The\nprojected range is defined as the average distance the muons travel in the\nz-direction (initial beam direction) before being stopped5.\n\nTo determine the projected range from the Geant4 simulation, we have to\ndefine at which energy we can consider the muon to be stopped. This can be\ndone by examining the average z-projection of the total momentum (pz/p)\nversus energy in Fig. 26. At high energies (> 1 keV) the muon momentum\npoints predominately in the +z-direction (pz/p ≈ 1), since the elastic scatter-\ning is rare at these energies. However, as the energy decreases, the pz/p also\ndecreases until it reaches 0 at around 100 eV. At this point, the muon direc-\ntion is completely randomized, i. e., the muon does not have any memory\nof its initial direction. Therefore, we can consider the muon to be stopped\nwhen its energy is below 100 eV.\n\nThis definition of “stopped muons” allows us to determine the projected\nrange of muons from the Geant4 simulation: we define the projected range\nto be the z-position of muons at 10 eV energy. The projected muon ranges\nobtained in this way are plotted in Figs. 27 and 28 for various initial muon\nenergies. The muon ranges extend from about 3 · 104 mm for 4 MeV muons,\ndown to only 2 mm for 10 keV muons (for 10 mbar of helium gas at 10 K). In\n\n5 Due to the elastic scattering, the projected range is in general smaller than the total path\nlength the muons travel.\n\n\n\n70 geant4 simulations of the muon stopping and drift in helium gas\n\nFigure 25: Stopping power for muons in He as a function of the muon en-\nergy in the laboratory frame, computed using Geant4 (black con-\ntinuous line) and electronic and nuclear stopping powers from\nSRIM [47] (dashed lines). The Geant4 simulation included the\nfollowing standard physics processes above 1 keV muon energy:\n\n“G4MuIonisation”, “G4MuMultipleScattering”, “G4MuBremsstrahlung” and\n“G4MuPairProduction”. Below 1 keV energy, the standard Geant4 pro-\ncesses have been turned off and the “custom” low-energy muon-He in-\nteractions were implemented, starting from the cross sections of Fig. 22.\n\nthe same figure the projected muon ranges calculated with SRIM are shown\nas vertical thick lines6. A good agreement between the range predicted by\nSRIM and the range obtained in our Geant4 simulation can be observed.\n\nWe tested the sensitivity of the range simulated with Geant4 to the stop-\nping power in the energy range between 300 eV and 1 keV by removing the\nadditional inelastic energy loss ∆Q of Eq. (180) from the Geant4 simulation\n(thus having only energy losses due to the charge exchange and elastic col-\nlisions). The simulated ranges with (solid lines) and without (dashed lines)\nthe additional energy loss ∆Q are shown in Fig. 28 for various initial muon\nenergies. We can see that the difference between the two cases is negligible.\nThe largest relative difference is obtained at low (initial) energy: for 10 keV\ninitial muon energy the range is shifted by about 0.5 mm.\n\nThus, we can conclude that the slowing-down of the muons from high\nenergy is not very sensitive to the exact implementation of the inelastic\nstopping power at energies below 1 keV. This is not surprising, because\nthe largest stopping power (Bragg peak) for muons in helium is at around\n10 keV and thus the most of the slowing-down happens in this energy re-\n\n6 To obtain the range from SRIM, we assumed the muon to be a light isotope of the proton\n(mµ = 1\n\n8.88mp). The SRIM software then calculates the range using the “projected range al-\ngorithm” (PRAL) [66], from the proton stopping power velocity-scaled to muons (see Fig. 20).\n\n\n\n3.2 muon range and slowing-down in he gas 71\n\nFigure 26: Plot of the average z-projection of the total momentum (pz/p) versus\nmuon energy for 104 muons of 4 MeV initial energy slowing down in\nHe gas (10 mbar at 10 K), obtained from the Geant4 simulation. The\nsmall distortion at 1 keV is caused by the switching from standard Geant4\nprocesses to our custom low-energy processes.\n\ngion. Therefore, our simplified implementation of the inelastic collisions at\nlow energies is adequate for our purpose and we do not expect to get signif-\nicantly different results by implementing every single possible inelastic pro-\ncess. Any differences between SRIM and the Geant4 ranges are thus more\nlikely caused by the difference in the elastic collision cross sections (stopping\npower). However, as explained in the previous section, we believe that our\nmodel of the elastic collisions, using the cross sections of [52], describes the\nmuon (proton) elastic collisions with helium atoms more accurately than the\nnuclear stopping power of SRIM.\n\n\n\n72 geant4 simulations of the muon stopping and drift in helium gas\n\nFigure 27: Distribution of the projected muon ranges for various initial muon ener-\ngies (in the laboratory frame) (colored histograms). The muon projected\nrange was defined to be the muon z-position at 10 eV energy. It is as-\nsumed that muons start at z = 0. The vertical lines show the correspond-\ning projected ranges obtained from SRIM.\n\nFigure 28: Distribution of the projected muon ranges for various initial muon en-\nergies (in the laboratory frame), obtained from Geant4 simulations with\n(solid colored histograms) and without (dashed colored histograms) ad-\nditional energy inelastic energy loss ∆Q of Eq. (180). The muon projected\nrange is defined to be the muon z-position at 10 eV energy. The vertical\nlines show the corresponding projected ranges obtained from SRIM.\n\n\n\n3.3 muon drift in an uniform electric field 73\n\n3.3 muon drift in an uniform electric field\n\nIn this section we will study the muon motion in helium gas with an uniform\nelectric field applied. This motion is the basis of the longitudinal compres-\nsion stage, as seen in Chapter 1. At these conditions, the muon energy gain\ndue to the electric field will compete with energy loss due to the collisions\nwith He gas atoms. If equilibrium between the two exists, the muon will\ndrift at a constant average velocity, the so-called drift velocity. The condition\nfor the equilibrium can be derived in an approximate way by averaging over\nmany collisions.\n\nSuch equilibrium is reached when the force produced by the electric field\nis equal to the average momentum loss per unit time:\n\neE = νc ·∆p, (181)\n\nwhere E is the applied electric field, ∆p is the average momentum loss per\ncollision and νc is the average collision frequency. When this condition\nis fulfilled, the muons drift at constant average velocity vd (drift velocity),\nwhich is parallel to E:\n\nvµ = vd. (182)\n\nNote that in general |vµ| > |vd|. The average muon kinetic energy is defined\nover vµ = |vµ| as usual:\n\nEµ =\n1\n\n2\nmµv2µ ≈\n\n1\n\n2\nmµv\n\n2\nµ. (183)\n\nFollowing the theoretical approach of [50, 67], as next, we would like to find\nthe average muon energy Eµ for an electric field of strength E applied inside\nthe gas of density N at equilibrium conditions. From this, the drift velocity\ncan also be obtained, as it is a function of the average muon kinetic energy:\nvd = vd(Eµ).\n\nWe start by evaluating the terms on the right-hand side in Eq. (181) at\nan energy Eµ7. The average momentum loss per collision at the energy Eµ Averaging over\n\nmany collisions at\nEµ energy is\nequivalent to\naveraging over the\ndistribution of the\nscattering angles at\nEµ energy.\n\n(averaging over many collisions at Eµ energy) is (see Appendix B):\n\n∆p ≈ ∆p(Eµ) = mrvd(Eµ)1− cosχ(Eµ) (184)\n\n= mrvd(Eµ)\nσMT (Eµ)\n\nσEL(Eµ)\n, (185)\n\nwhere mr =\nmµmHe\n\nmµ+mHe\nis the reduced mass and χ is scattering angle in the\n\ncenter-of-mass frame. In Eq. (185), we have expressed the average 1− cosχ\n\n7 In principle the average momentum loss per collision should be obtained by also averaging\nover the muon energy distribution f(Eµ): ∆p = 1\n\nEµ\n\n∫\n∆p(Eµ)f(Eµ)dEµ and not just over the\n\nscattering angle distribution. Since the energy distribution is difficult to obtain, we make an\napproximation ∆p = ∆p(Eµ) (the bar over ∆p on the right hand side represents averaging\nover many collisions, i. e., over scattering angles). Similar approximations are also made for\nother variables, such as collision frequency and energy loss per collision.\n\n\n\n74 geant4 simulations of the muon stopping and drift in helium gas\n\nusing the definition of the momentum-transfer cross section of Eq. (91).\nThe average collision frequency is given by\n\nνc ≈ νc(Eµ) = NvrσEL(Eµ) = NvµσEL(Eµ), (186)\n\nwhere N is the gas density, vr is the average relative velocity of muon and\nHe atom and σMT is the momentum-transfer cross section. In the second\nequation we have assumed vr = vµ, i. e., that the He gas atoms are at rest.\n\nBy inserting Eqs. (185) and (186) into Eq. (181) we get:\n\neE ≈ mrvdNvµσMT (Eµ). (187)\n\nSince E and vd are parallel, from now on we will drop vectors. Furthermore,\nwithout loss of generality, we assume in the following that the electric field\npoints in the z-direction.\n\nWe still need to relate vd and vµ to obtain the condition for the equilibrium\ndrift. This can be done by comparing the average momentum change ∆p per\ncollision with the average energy loss ∆E per collision, which are related by\nthe following expression:\n\nνc(Eµ) ·∆p(Eµ) =\n∆E(Eµ)\n\n∆z(Eµ)\n, (188)\n\nwhere ∆z is the average distance the muon travels in the z-direction between\ntwo collisions 8. The average distance that the muon of energy Eµ travels in\nthe z-direction between two collisions is given by:\n\n∆z(Eµ) = vd(Eµ)τ(Eµ), (189)\n\nwhere τ is mean free time between two collisions.\nThe average energy loss in an elastic collision can be evaluated from Eq. (44):\n\n∆E(Eµ) = 2\nmµmHe\n\n(mµ +mHe)2\nEµ(1− cosχ). (190)\n\nWith that Eq. (188) becomes:\n\nνc(Eµ)mrvd(Eµ)1− cosχ(Eµ) = 2\nmµmHe\n\n(mµ +mHe)2\nEµ\n1− cosχ(Eµ)\nvd(Eµ)τ(Eµ)\n\n(191)\n\nv2d(Eµ) =\n2\n\nmr\n\nmµmHe\n\n(mµ +mHe)2\nEµ (192)\n\nv2d(Eµ) =\n2\n\nmµ +mHe\nEµ, (193)\n\n8 Note that the term on the right hand side of Eq. (188) looks similar to the stopping power, as\ndefined by Eq. (25) in Chapter 2, but it is not the same. The stopping power is the average\nenergy loss per unit path length. The muon trajectory at low energy is however not straight\ndue to the elastic scattering and thus the average distance that muon travels in the z-direction\nbetween two collisions is in general smaller than the mean free path of the muon. Therefore,\n∆E\n∆z\n\nis larger than the stopping power (along the path) at same energy.\n\n\n\n3.3 muon drift in an uniform electric field 75\n\nwhere we used τ = 1/νc. Expressing the kinetic energy Eµ = 1\n2mµv\n\n2\nµ in\n\nterms of the average muon velocity vµ we find a relation between drift ve-\nlocity and vµ:\n\nvd(vµ) =\n\n√\nmµ\n\nmµ +mHe\nvµ. (194)\n\nInserting Eq. (194) into Eq. (187) we get:\n\ne\nE\n\nN\n≈ mr\n\n√\nmµ\n\nmµ +mHe\nv2µσMT (Eµ) (195)\n\ne\nE\n\nN\n≈ 2\n√\n\nmµ\n\nmµ +mHe\n\nmHe\nmµ +mHe\n\nEµσMT (Eµ) (196)\n\ne\nE\n\nN\n≈ 0.32 · EµσMT (Eµ), (197)\n\nwhere in the last line we have inserted the values for muon and helium\nmasses.\n\nIf we want to express the mass density ρ of He (in g/cm3) instead of the\nnumber density N (in cm−3) the Eq. (197) has to be modified as:\n\ne\nE\n\nρ\n≈ 0.48 · 1023g−1 · EµσMT (Eµ) (198)\n\nwhere we inserted N = NA\nM ρ, with NA = 6.022 · 1023 mol−1 being the Avo-\n\ngadro number and M = 4 g/mol the molar mass of He.\nThe right-hand side of this equation is the average stopping power along\n\nthe z-direction. The plot of this stopping power versus the muon kinetic\nenergy is shown in the Fig. 29 (using the momentum-transfer cross section\nof Fig. 22). The equilibrium muon energy can be determined graphically\nfrom this plot by finding the intersection of eE/ρ with the average stopping\npower along the z-direction. The example for eE/ρ = 25 MeVcm2/g, shown\nin Fig. 29, corresponds to an average equilibrium kinetic energy of 0.28 eV.\n\n\n\n76 geant4 simulations of the muon stopping and drift in helium gas\n\nFigure 29: Average stopping power along the z-direction, defined by the right-hand\nside of Eq. (198), versus the muon energy in laboratory reference sys-\ntem. The momentum-transfer cross section of Fig. 22 has been used (af-\nter transforming the center-of-mass energy to energy in the laboratory\nframe using Eq. (40)). The equilibrium muon energy can be determined\ngraphically from this plot by finding the intersection of eE/ρ with the\naverage stopping power along the z-direction: two examples are shown\nfor eE/ρ = 25 MeVcm2/g and for eE/ρ = 45 MeVcm2/g (solid horizontal\nlines). The resulting average equilibrium energies are indicated by the\nvertical dashed lines.\n\nWe can see that the maximum stopping power is about 50 MeVcm2/g.\nTherefore, for large enough reduced electric field strengths (E/ρ), the stop-\nping power due to the elastic collisions with the helium gas is smaller than\nthe energy gained from the electric field, so that the muon will be accelerated.\nThe acceleration continues until the muon energy becomes large enough so\nthat also the inelastic collisions start contributing to the stopping power (at\nseveral hundred eV). This sudden increase of the muon drift velocity with in-\ncreasing reduced electric field strength is the so-called runaway effect, which\nwas first predicted theoretically for protons and deuterons in helium by [18]\nand confirmed experimentally in [68]. The runaway condition for the muons\nin helium can be written as:\n\ne\nE\n\nρ\n> 0.48 · 1023g−1 · EµσMT (Eµ). (199)\n\nThe onset of the runaway is important for the longitudinal compression\nstage, where it allows a rapid muon compression in the z-direction, reducing\nthe muon losses (due to the short muon lifetime), as explained in Chapter 1.\n\nThe onset of the runaway for muons in helium was studied using Geant4\nsimulations. In the simulation, 104 positive muons were generated with\n\n\n\n3.3 muon drift in an uniform electric field 77\n\n0.1 eV energy inside a large helium volume of 5 mbar pressure and 293 K\ntemperature, corresponding to a density of 8.2152 · 10−7 g/cm3. An uniform\nelectric field in the z-direction was applied inside the helium volume. The\nfield strength was varied to achieve different values of eE/ρ, ranging from\n20 to 80 MeVcm2/g.\n\nThe time evolution of the kinetic energy of a single muon, for eE/ρ =\n\n25 MeVcm2/g is shown in Fig. 30. The muon has initial energy of 0.1 eV, but\nwithin 100 ns it gains energy from the applied electric field until its energy\nreaches approximately 0.3 eV. After that, the muon energy oscillates around\nthis equilibrium value. The average equilibrium energy at these conditions,\ndetermined from Eq. (198), is marked with a vertical dashed line in Fig. 30.\nWe see that this equation gives a good prediction of the equilibrium muon\nenergy, thus indirectly validating the Geant4 simulation to the theory of [18].\n\nFigure 30: Time evolution of the kinetic energy of a single muon in He gas, for\neE/ρ = 25 MeVcm2/g, simulated with Geant4. The dashed line indicates\nthe average equilibrium energy calculated from Eq. (198).\n\n\n\n78 geant4 simulations of the muon stopping and drift in helium gas\n\nFigure 31: Time evolution of the kinetic energy of a single muon in He gas, for\neE/ρ = 45 MeVcm2/g, simulated with Geant4. The dashed lines indicate\nthe average equilibrium energies calculated from Eq. (198).\n\nFor a larger reduced electric field strength, eE/ρ = 45 MeVcm2/g, the\nevolution of the muon energy with time has a different behavior, as shown\nin Fig. 31. In this case, at late times (after approximately 100 ns) the muon\nenergy oscillates between two different “average” values, 0.7 and 30 eV ap-\nproximately. This is again consistent with the prediction of the Eq. (198),\nsince in this case, the line eE/ρ = 45 MeVcm2/g intersects the stopping\npower curve at four energies, as shown in Fig. 29, corresponding to average\nequilibrium values of 0.64 and 33.8 eV9. These two average equilibrium en-\nergies are shown as black dashed lines in Fig. 31. Also in this case we get\na good agreement between the equilibrium energies obtained from Geant4\nsimulation and the energies predicted by the Eq. (198).\n\n9 The intersection with a negative stopping-power slope are unstable: the muon is accelerated\nto higher energies.\n\n\n\n3.3 muon drift in an uniform electric field 79\n\nFigure 32: Muon energy distributions at equilibrium for several values of eE/ρ, sim-\nulated with Geant4. The vertical dashed lines indicate the average equi-\nlibrium energies predicted by Eq. (198).\n\nThe muon energy distribution at a given time can be obtained by simu-\nlating the trajectories of a large number of muons in the uniform electric\nfield. The energy distribution of muons at t = 2000 ns for several values\nof eE/ρ (below the maximum of the stopping power, under which the equi-\nlibrium is possible) is shown in Fig. 32. The dashed lines again mark the\nequilibrium energies predicted by Eq. (198). These energies are also listed in\n\n\n\n80 geant4 simulations of the muon stopping and drift in helium gas\n\nthe Table 1 for various values of eE/ρ, along with the corresponding electric\nfield strengths at the He gas density of 8.2152 · 10−7 g/cm3.\n\neE/ρ E equilibrium energies\n\n(MeVcm2/g) (V/cm) (eV)\n\n20 16.4 0.21 -\n\n25 20.5 0.28 -\n\n30 24.6 0.36 -\n\n35 28.8 0.52 12.93\n\n40 32.9 0.58 21.38\n\n45 37.0 0.64 33.77\n\n50 41.1 0.78 58.50\n\nTable 1: Muon average equilibrium energies in the laboratory frame for various val-\nues of eE/ρ . The second column shows the corresponding electric field\nstrength for a He gas density of 8.2152 · 10−7 g/cm3.\n\nFrom this table we see that for eE/ρ < 35 MeVcm2/g we expect to have\nonly one equilibrium energy, while for eE/ρ > 35 MeVcm2/g there are two\nstable values of the drift velocity. This is consistent with the distributions\nof Fig. 32, simulated with Geant4, showing one or two maxima, depending\non the eE/ρ value. The positions of the maxima are in relatively good agree-\nment with the average equilibrium energies predicted by Eq. (198). There\nseems to exist a small systematic offset towards higher energies in the simu-\nlated energy distributions compared to the predictions of Eq. (198). This is\nnot so surprising, because the Eq. (198) gives only an approximate descrip-\ntion of the equilibrium.\n\nIt is also interesting to consider what happens at eE/ρ above the maximum\nof the stopping power curve, i.e., above roughly 50 MeVcm2/g. The muon\ndistributions at t = 2000 ns for eE/ρ up to 80 MeVcm2/g are shown in\nFig. 33. We can see again that at low eE/ρ there is only one equilibrium\nenergy (below 1 eV). As eE/ρ is increased, the muons start crossing the first\n“bump” of the stopping power curve (at 1 eV), resulting in two equilibrium\nenergies. With a further increase of the eE/ρ value, an increasing number of\nmuons accumulates at the second (higher) equilibrium energy. For eE/ρ >\n50 MeVcm2/g the muons cross also the second “bump” of the stopping\npower curve (at 100 eV) and thus the energy distribution shifts to much\nhigher values, determined by the intersection of the eE/ρ with the inelastic\nstopping power curve (and not the by elastic one).\n\n\n\n3.4 muon drift in crossed electric and magnetic fields 81\n\nFigure 33: Muon energy distributions at equilibrium for several values of eE/ρ up\nto 80 MeV cm2/g, simulated with Geant4. Depending on the eE/ρ value,\nwe can observe one or two maxima. For eE/ρ > 50 MeV cm2/g, the ma-\njority of muons are at “high-energy” (several hundred eV), thus demon-\nstrating the runaway effect (see text).\n\n3.4 muon drift in crossed electric and magnetic fields\n\nIn this section we will study the muon motion in helium gas in crossed\nelectric and magnetic fields, which is the basis of the transverse compression\nstage. As explained in Chapter 1, the muon drift direction in crossed electric\nand magnetic fields depends on the elastic collision frequency with helium\ngas atoms as:\n\ntan ϑ =\nνc\n\nω\n, (200)\n\nwhere ϑ is the angle between the muon drift direction and the Ê× B̂-direction,\nνc is the average collision frequency and ω is the muon cyclotron frequency.\nThe deflection angle ϑ increases with the collision frequency νc, which is\ngiven by:\n\nνc(vr) = NσMT (vr)vr, (201)\n\nwhere N is the helium gas density, σMT is the momentum-transfer10 cross\nsection for muon-He elastic collisions and vr is the relative velocity between\nmuons and He atoms. If we take He atoms to be at rest we can write the\ncollision frequency as:\n\nνc(vµ) = NσMT (vµ)vµ = NσMT (Eµ)vµ. (202)\n\n10 We use here the momentum-transfer cross section and not the total elastic cross section, since\nonly collisions that change significantly the muon direction will have impact on the muon\ndrift direction.\n\n\n\n82 geant4 simulations of the muon stopping and drift in helium gas\n\nSince the momentum-transfer cross section depends on the muon energy\n(see Fig. 22), the collision frequency is also energy-dependent11. This implies\nthat the collision frequency is a function, not just of He gas density, but also\nof the electric field strength - since the muon energy depends on the applied\nelectric field. Therefore, the muon drift direction is a function of the He gas\ndensity and the electric field strength as well.\n\nIn the following, the dependence of the average muon drift angle ϑ on\nthe helium gas density and electric field strength, i. e., on the collision fre-\nquency, is studied using the Geant4 simulation described in Section 3.1. In\nthe Geant4 simulation presented here, a 5 T magnetic field pointing in +z-\ndirection and an uniform electric field pointing in +y-direction are applied\ninside a large He volume of 8 mbar pressure. The trajectories of 20 posi-\ntive muons of 1 eV initial energy were simulated for electric field strengths\nranging from 0.5 to 2.0 kV/cm and for He temperatures from 4 K to 300 K.\n\nFigure 34: Simulated muon trajectories (projected in the yx-plane) for an electric\nfield of strength E = 1.2 kV/cm pointing in the +y-direction and 5 tesla\nmagnetic field pointing in the +z-direction, for various He gas temper-\natures (at 8 mbar pressure). The muons are initially generated at the\norigin of the coordinate system.\n\nSimulated muon trajectories at a fixed electric field strength E = 1.2 kV/cm\nare shown in Fig. 34 for several He gas temperatures. At t = 0, muons are at\nthe origin of the coordinate system (x,y, z = 0). Under the influence of the\napplied electric and magnetic fields, the muons drift in the yx-plane, form-\n\n11 Note that in principle Eq. (202) should be averaged over the energy distributions of the\nmuons drifting in the gas [16]. Thus, Eq. (200) should be written as:\n\ntan ϑ =\nνc\n\nω\n, (203)\n\nwhere νc is the collision frequency averaged over the muon energy distribution at a certain\nelectric and magnetic field strengths and at a certain gas density.\n\n\n\n3.4 muon drift in crossed electric and magnetic fields 83\n\ning a certain angle with the x-axis. We can notice that the average angle ϑ\nwith the respect to the x-axis (Ê× B̂-direction) decreases with increasing He\ngas temperature. This is consistent with Eq (201), since larger temperatures\ncorrespond to lower He gas densities, implying lower collision frequencies\nand thus smaller drift angles.\n\nFigure 35: Simulated muon trajectories (projected in the yx-plane) for a fixed He gas\ndensity (8 mbar at 12 K) and for various electric field strengths (pointing\nin the +y-direction). The magnetic field of 5 tesla strength is pointing in\nthe +z-direction. The muons are initially generated at the origin of the\ncoordinate system. Note that the trajectories for 0.5, 0.7 and 0.9 kV/cm\nlargely overlap.\n\nAs next, we study what happens when we vary the electric field strength,\nwhile keeping the He gas temperature fixed. The simulated muon trajecto-\nries in 8 mbar of He gas at 12 K are shown in Fig. 35 for several electric\nfield strengths. At electric field strengths below 0.9 kV/cm, the average drift\nangle ϑ is large and roughly independent of the field strength. Above 0.9\nkV/cm, the average muon drift angle ϑ decreases with increasing electric\nfield, implying lower collision frequency at higher electric fields.\n\n\n\n84 geant4 simulations of the muon stopping and drift in helium gas\n\nFigure 36: Average muon collision frequency in He gas at 8 mbar pressure and at\n12 K temperature as a function of the muon kinetic energy. The collision\nfrequency was calculated using Eq. (202) and the momentum-transfer\ncross section of Fig. 22 (after transforming the energy from the center-of-\nmass to the laboratory frame).\n\nSuch a behavior of the muon drift angle can be understood by studying\nthe dependence of the collision frequency of Eq. (202) on the muon energy.\nA plot of the collision frequency as a function of the muon energy in He\ngas of 8 mbar at 12 K is shown in Fig. 36 (obtained using Eq. (202) and\nthe momentum-transfer cross section of Fig. 22). At muon energies below\n1 eV the collision frequency is almost independent of energy. For energies\nabove 1 eV the collision frequency decreases rapidly with increasing muon\nenergy. Therefore, stronger electric fields, leading to larger average muon\nenergies, result in smaller collision frequencies and accordingly in smaller\ndrift angles.\n\nTo relate the collision frequency to the simulated muon drift angle, we\nexamine the muon energy distribution for various field strengths, as shown\nin Fig. 37. For electric field strengths below 0.9 kV/cm, the muon energy\nis mostly below 1 eV. This implies that at these field strengths the collision\nfrequency is roughly independent of the energy (see Fig. 36), so that the drift\ndirection is roughly independent of the field strength. This is consistent with\nthe trajectories plotted in Fig. 35. For field strengths above 0.9 kV/cm, the\nmuon energy is (mostly) larger than 1 eV, so that an increase of the electric\nfield strength results in a decrease of the collision frequency and drift angle,\nas visible from the muon trajectories plot of Fig. 35.\n\n\n\n3.5 conclusions 85\n\nFigure 37: Muon energy distributions for various electric field strengths in He at\n8 mbar pressure and 12 K temperature (corresponding to the trajecto-\nries of Fig. 35). The energy distribution is obtained by saving the muon\nenergy at each simulation step (at all times).\n\nAs a summary, in Fig. 38 we show the simulated muon drift angle ϑ as\na function of the electric field strength, for He gas temperatures ranging\nfrom 4 to 300 K. The obtained drift angles are consistent with the discussion\nabove: at a fixed electric field strength the average angle ϑ increases with\nHe gas density (decreasing temperature). At a high gas density (low tem-\nperature), the drift angle is roughly independent of the applied electric field\nstrength for all the simulated electric fields (0.5-2.0 kV/cm). As we increase\nthe temperature, this behavior changes: the drift angle is only constant up\nto a certain (threshold) electric field strength, while above this threshold the\ndrift angle decreases with increasing energy. This threshold electric field\ndecreases with increasing gas temperature (decreasing gas density). We can\ninterpret this threshold electric field as the field needed to have a muon\nmean kinetic energy of 1 eV.\n\n3.5 conclusions\n\nThis chapter was concerned with the Monte-Carlo simulation of muon stop-\nping and drift in He gas, needed for the correct modeling and optimization\nof the muCool device. The simulations were done using the Geant4 toolkit.\nIt was demonstrated that the standard Geant4 processes for muons are in-\nadequate for describing low-energy muon-He interactions. Due to this limi-\ntation, the low-energy muon-He elastic collisions and charge exchange had\nto be manually implemented in Geant4. Such implementation requires the\nknowledge of the various cross sections and the details of the collision dy-\nnamics. For this purpose, we made use of the findings of Chapter 2 regard-\ning the collision kinematics and the scaling of proton-He to muon-He cross\n\n\n\n86 geant4 simulations of the muon stopping and drift in helium gas\n\nFigure 38: Average muon drift angle ϑ as a function of the electric field strength, for\nvarious He gas temperatures (at 8 mbar pressure), obtained from Geant4\nsimulations.\n\nsections and stopping power. A short outline of the various steps needed\nto simulate the muon trajectories was presented, with emphasis on how to\ndeal with multiple “competing” processes.\n\nSuch a “custom” Geant4 simulation was then used to study the muon stop-\nping and drift in several simple scenarios, which are the foundation of the\nmuCool device: slowing-down of high-energy muons in He gas, muon drift\nin an uniform electric field and muon drift in crossed electric and magnetic\nfields. These studies allowed us to partially validate our Geant4 simula-\ntion by comparing the results of the simulation with available SRIM range\ndata [47] and with the theoretical model of proton mobility of Ref. [18].\n\nFurthermore, these simulations allowed us to quantify the requirements\nfor the muCool device in terms of the applied electric fields strengths and\ngas densities. For muons drifting in an uniform electric field, which is the\nbasis of the longitudinal compression stage, a rapid increase of the muon\n\n\n\n3.5 conclusions 87\n\ndrift velocity was observed for reduced electric fields E/ρ & 50 MVcm2/g.\nAt such conditions, a fast longitudinal compression can be achieved, thus\nminimizing muon losses due to a short muon lifetime.\n\nRegarding the muon drift in crossed electric and magnetic fields (corre-\nsponding to the field arrangement in the transverse compression stage) the\nGeant4 simulation brought out the fact that the muon drift direction de-\npends strongly, not just on the He gas density, but also on the applied elec-\ntric field strength. Such a behavior is explained by the energy-dependence\nof the elastic collision cross sections (and thus of the collision frequency).\nThe main takeaway is that there exists a threshold electric field strength\n(which depends on the gas density), below which the muon drift is pre-\ndominantly in the electric field direction. The muon drift in the predomi-\nnantly Ê× B̂-direction is only possible if the electric field strength exceeds\nthis threshold value. As explained in Chapter 1, in the transverse compres-\nsion stage the muons stopped in the upper part of the target should drift\nin the Ê× B̂-direction, otherwise they will simply crash into the top wall of\nthe target. Thus, practically, the transverse compression is possible only for\nstrong enough electric fields.\n\n\n\n\n\n4\nT R A N S V E R S E C O M P R E S S I O N\n\nThe transverse compression stage is the first stage of the muCool device. In\nthis stage the secondary muon beam is stopped in the cryogenic helium gas,\nthereby reducing the muon energy from few MeV to few eV. Subsequently,\nthe transverse (vertical) size of the muon beam is reduced to below 1 mm\nsize using a combination of crossed electric and magnetic fields and gas\ndensity gradients.\n\nThe first engineering run for the transverse compression stage was at-\ntempted during a beam-time in 2014. Several aspects of the setup were\nsuccessfully tested, such as muon beam coupling to the setup, positron de-\ntection system at cryogenic temperatures and the cryostat. Unfortunately,\ndue to a helium leak from the transverse target at cryogenic temperatures\nand the breaking of the electrical contacts during the cool-down, we were\nnot able to test the transverse compression stage.\n\nIn 2015 the effort was invested in solving the aforementioned problems\nwith the transverse target. The target design was optimized to prevent the\nhigh voltage breakdown when applying high electric fields in the helium\ngas. In another beam-time in December 2015, we successfully demonstrated\nthe transverse compression for the first time.\n\nThis chapter is divided into three sections. The first section provides an\noverview of the experimental setup used to demonstrate the transverse com-\npression, including the description of the target construction, the detection\nsystem and the beam injection. The second section presents the measure-\nments that demonstrated the transverse compression. Comparison with the\nGeant4 simulations is also given. Finally, in the third section, a broader\nstudy of the muon motion as a function of the applied electric field is pre-\nsented, which expands further the discussion of the muon drift dependence\non energy from Chapter 3.\n\n4.1 setup for the transverse compression\n\nTo achieve transverse compression of the muon beam, several \"ingredients\"\nare needed, as explained in Chapter 1:\n\n• a helium gas with a density gradient at cryogenic temperatures,\n\n• an electric field at 45° angle, of 1.5 kV/cm strength,\n\n• a 5 Tesla magnetic field.\n\nThese requirements make the target construction challenging and limit\nthe materials that can be used. The target design that arises from these\nconstraints is described in the following subsection.\n\n89\n\n\n\n90 transverse compression\n\nTransverse target design and materials The gas volume of the transverse\ntarget with dimensions 40× 23.5× 250 mm3 is enclosed by a 50 µm Kapton\nfoil, which is folded around the triangular PVC end-caps (see Fig. 39). Both\nthe plastic end-caps and the Kapton foil are electric insulators with a low\nthermal conductivity. These properties are necessary in order to reduce the\nheat exchange between the top and the bottom of the target and to sustain\nhigh voltage. Sapphire plates glued to the top and the bottom of the target\ndefine homogeneous temperatures of 18.6 K and 6 K, at the top and the bot-\ntom of the target, respectively. Single crystal sapphire was chosen because\nit has a large thermal conductivity at low temperatures and it is electrically\ninsulating.\n\nFigure 39: Photo of the realized transverse compression target. The gas volume of\nthe target is enclosed by the Kapton foil, lined with the electrodes which\ndefine the required electric field. The applied voltages, that generate an\nelectric field of 1.4 kV/cm are also indicated. Also visible in the photo\nare the top and the bottom sapphire plates, used to define the vertical\ntemperature gradient inside the target.\n\nCare must be taken of how the various materials are assembled together\ndue to different coefficients of thermal expansion: for example, sapphire\nand plastic cannot be directly glued together. Therefore, in our target, the\nsapphire plates rest only on two additional Al2O3 supports inside the target,\nwithout being directly glued to the plastic end-caps. All components are\nglued with Stycast 1266 glue, which works at cryogenic temperatures. The\noverview of the relevant thermal and electrical properties of these materials\nat low temperatures along with the references are given in Table 2.\n\nTemperature gradient The required temperature gradient is produced\nby thermally contacting the bottom sapphire plate to a copper cold finger\n(Fig. 40) while the top sapphire is heated with approximately 500 mW power.\nThe cold finger is made of a 1 m long copper bar coupled to a pulse tube\ncryostat with 1.5 W cooling power (at 4 K). In this way, a temperature gradi-\n\n\n\n4.1 setup for the transverse compression 91\n\nMaterial\nThermal conductivity at 4 K Integral thermal contraction (∆L/L)\n\n(W/cm·K) from 293 K to 4 K\n\nKapton foil 1.1 · 10−4 [69] 4.4 · 10−3 [69]\n\nSapphire 2.30 [70] 6− 7 · 10−4 [71]\n\nAl2O3 4.9 · 10−3 [70] −\n\nPVC 2.7 · 10−4 [70] 1 · 10−2 [72]\n\nTable 2: Thermal conductivity and integral thermal contraction of the main materi-\nals used for the construction of the transverse compression target.\n\nFigure 40: Sketch of the setup used in the 2015 beam-time to measure the muon\ntransverse compression. The transverse compression target is mounted\non the “cold finger” made of a 1 m long copper bar. The cold finger is\ncoupled to the second cooling stage of the pulse tube cryocooler with\n1.5 W of cooling power (at 4 K). The target mounted on the cold finger\nis placed in the center of the 5 T magnet with a bore hole of 20 cm di-\nameter. The cryostat vacuum tube and the thermal shield placed inside\nthe cryostat are also indicated. To test the transverse compression, about\n2 · 104 µ+/s at 12.5 MeV/c momentum were injected into the target. Be-\nfore entering the target the muons have to pass through several thin foils,\nas shown in the sketch: a 55 µm thick entrance detector, a 25 µm Kap-\nton foil (sealing the cryostat vacuum), a 2 µm Mylar foil coated with a\n0.2 µm-thick aluminum layer (acting as a thermal shield) and finally a\n12 µm Kapton foil of the target window.\n\nent from 6.0 K to 18.6 K has been achieved. The ability to define the required\ntemperature gradient was confirmed experimentally using neutron radiog-\nraphy on 3He targets [22].\n\n\n\n92 transverse compression\n\nFigure 41: COMSOL Multiphysics® simulation of the temperature distribution in\nthe transverse compression target (color scale). The temperatures of the\nbottom and top sapphire plates were fixed in the simulation, matching\nthe experimentally obtained values of 6 K and 18.6 K, respectively.\n\nThe gas density distribution was simulated using COMSOL Multiphysics®,\ntaking into account the temperature-dependent thermal conductivities of the\nvarious materials used in the target construction and of the helium gas. The\nresults of the simulation are shown in Fig. 41. In the simulation, the tem-\nperatures of the bottom and top sapphire plates were fixed, matching the\nexperimentally obtained values of 6 K and 18.6 K, respectively.\n\nElectric fields The Kapton foil enclosing the gas volume is lined with\nthin metallic electrodes. The electric field is defined by applying appropriate\nvoltages to six of these electrodes, in the target corners, which are connected\nto the rest of the electrodes via SMD resistors of 100 MΩ resistance, acting as\na voltage divider. Typical values of the applied voltage are shown in Fig. 42.\nThese values were optimized with COMSOL Multiphysics® to obtain the\nelectric field in the (1, 1, 0) direction, and as uniform as possible. The shape\nof the obtained electric field and the equipotential lines calculated by Comsol\nare shown in Fig. 42.\n\nBeam injection To test the transverse compression about 2 · 104 µ+/s\nat 12.5 MeV/c momentum were injected into the target, which is placed in\nthe center of the 5 Tesla solenoid bore hole (see Fig. 40). Before entering\nthe target, muons have to pass through a 55 µm thick entrance detector\nand three different thin foils: 25 µm Kapton, sealing the cryostat vacuum,\n\n\n\n4.1 setup for the transverse compression 93\n\nFigure 42: COMSOL Multiphysics® simulation of the electric potential (color scale)\nin the transverse compression target, producing an electric field (black\nlines) of 1.4 kV/cm on average at 45° with respect to the x-axis. The po-\ntential is defined by the electrodes at the target walls (gray rectangles, not\nto scale). The voltage is applied only to few electrodes (blue rectangles),\nwhile on the rest of the electrodes the potential is linearly distributed via\n100 MΩ resistors, acting as voltage dividers.\n\n2 µm Mylar foil coated with a 0.2 µm-thick aluminum layer, acting as a\nthermal and light shield and finally a 12 µm Kapton foil enclosing the He\ngas target (target window). A copper collimator in front of the target defines\nthe entrance position of the muon beam in the target.\n\nDetectors Several detectors were placed around the target, as shown\nin Fig. 43, to monitor the movement of the muon swarm by detecting the\npositrons from µ+ decays. The detectors above the top sapphire (A1 - A3,\nA8f) were mounted inside a brass collimator in order to improve the posi-\ntion resolution of these detectors by collimating the positrons that do not\noriginate from the µ+ decay in the vicinity of the scintillator. The majority\nof the detectors were plastic scintillator bars with a groove, inside which\na wavelength-shifting fiber was glued. These 2 m-long wavelength shift-\ning fibers transported the scintillating light produced by the scintillators\nplaced at cryogenic temperatures to room temperature. The scintillator bars\nwere wrapped with several layers of Teflon tape to improve the scintillation\nlight collection into the wavelength-shifting fiber. The remaining detectors\n(A2f1, A4f2 and A8f2) were plastic scintillating fibers, which have a smaller\ncross section and thus a better position resolution than the scintillator bars,\nbut smaller total detection efficiency. Both wavelength-shifting fibers and\nscintillating fibers were read out by Silicon Photomultipliers (SiPM) with\n1.3× 1.3 mm2 active detection area.\n\n1 Round fiber of 1.5 mm diameter.\n2 Square 1.5× 1.5 mm2 fiber.\n\n\n\n94 transverse compression\n\nFigure 43: Sketch of the detector setup used to test the transverse compression in the\n2015 beam-time. Several plastic scintillators and scintillating fibers were\nplaced around the transverse compression target to detect the positrons\nfrom muon decays. Different detectors are sensitive to different muon de-\ncay positions inside the target, thus enabling us to monitor the movement\nof the muon swarm.\n\nThe probability to detect the positron from the muon decay as a function\nof the muon decay position is shown in Fig. 44 for the various detectors.\nDifferent detectors are sensitive to different regions inside the target. By\nrecording the time-spectra for each detector (number of counts as a function\nof the time difference between the hit in the corresponding detector and in\nthe entrance detector), we can deduce the average position evolution of the\nmuon ensemble versus time.\n\n4.2 results of the 2015 beam-time\n\nIn 2015 we performed two sets of measurements with various muon beam\ncollimations (see Fig. 45):\n\n• with a big aperture\n\n• with two smaller apertures.\n\nWith the big aperture we optimized the injected muon rate, while the smaller\napertures allowed us to resolve finer details of the muon motion.\n\nbig aperture\n\nMuon trajectories - compression The simulation of the muon trajectories\nobtained by applying the 5 Tesla magnetic field, the electric fields of Fig. 42\n\n\n\n4.2 results of the 2015 beam-time 95\n\nFigure 44: Projections of the acceptance of the A1, A2R, A2L and A3 positron de-\ntectors in the xy-plane. In each acceptance plot the position of the corre-\nsponding detector is sketched (blue rectangle).\n\nFigure 45: Sketch of the transverse target highlighting the two muon beam collima-\ntion types used during the data taking: a single large aperture (left) and\ntwo smaller apertures, the so-called “top hole” and “bottom hole” (right).\n\nand the temperature gradient of Fig. 41 in 8.6 mbar of He gas is shown in\nFig. 46 (left). Muons are stopped in the He gas at around x = −15 mm and\ndrift in the +x-direction, while simultaneously compressing in y-direction.\nThe muon swarm size in the y-direction is decreased from the initial value\nof 10 mm to about 0.7 mm size (FWHM) at x = 15 mm, while the energy is\ndecreased from the initial mean value of 830 keV with about 17 keV spread\nto about 5 eV with 5 eV spread, as shown in Fig. 47. Note that for x >\n15 mm the muon trajectories bend upwards (in +y-direction) because, at\nthis position, the electric field and the temperature gradient are not nicely\ndefined anymore: both electrodes and sapphires end at that position.\n\nTime-spectra - compression The resulting time-spectra under such con-\nditions are presented in Fig. 48 (dots and red lines). The number of counts in\neach detector was corrected for the muon decay by multiplying the counts\nwith exp(t/2198 ns). The number of counts in detector A1 increases with\nthe time until about 2500 ns. This indicates that the muons were gradually\n\n\n\n96 transverse compression\n\nFigure 46: (Left) Projections of the muon trajectories in the yx-plane for the big\naperture. The trajectories were simulated in Geant4 using the simulated\nelectric field of Fig. 42 and the simulated density gradient of Fig. 41 (6-\n18.6 K at 8.6 mbar) within a 5 tesla magnetic field. The 13.4 MeV/c\nmuon start in front of entrance detector and after passing through sev-\neral foils (see Fig. 40) they are stopped in the transverse compression\ntarget at around x = −15 mm. After that, the muons start drifting in the\n+x-direction, while simultaneously being compressed in the y-direction.\nWithin ≈ 3 µs they reach the tip of the target at x = 15 mm, where their\nspread has been reduced to only 0.7 mm in the y-direction.\n(Right) Projections of the muon trajectories in the yx-plane for the big\naperture and “pure drift” density conditions, i. e., without density gradi-\nent (3.5 mbar and 5 K on average). In this case, the muons also drift in\nthe +x-direction, but without compressing in the y-direction.\n\nFigure 47: (Left) Initial (black) and final (red) y-distribution of the muons under the\n“compression” conditions (corresponding to the trajectory simulation of\nFig. 46 (left)). The final distribution is evaluated at x = 15 mm.\n(Right) Initial (black) and final (red) muon energy distribution under the\n“compression” conditions (corresponding to the trajectory simulation of\nFig. 46 (left)). The inset shows a zoomed plot of the final muon energy\ndistribution using a log-scale for the energy axis.\n\n\n\n4.2 results of the 2015 beam-time 97\n\nmoving closer towards the tip of the target, where the A1 detection efficiency\nis higher, i. e., the acceptance region of A1 (see Fig. 44). After about 2500 ns\nthe number of counts does not change with the time, which tells us that the\nmuons were not moving anymore, i. e., they reached the target walls. On\nthe other hand, the number of counts in the detector A3 first rises with time\nand then it decreases, which tells us that the muons first entered and then\nexited the acceptance region of the detector (they \"flew by\" the detector).\n\nFigure 48: Measured time spectra of the detectors A1 and A3 for the big aperture\nsetting and under the “compression” (red dots and lines) and “pure drift”\n(solid black dots and lines) conditions. In both cases the maximum ap-\nplied HV was 5 kV. These measurements correspond to the Geant4 simu-\nlations of Fig. 46. The black dashed lines are the “pure drift” time spectra\nscaled to have the the same number of counts as the “compression” time-\nspectra (red lines).\n\nThroughout this\nchapter, the term\n\"compression\" will\nbe used to refer to\nthe case with a\nvertical density\ngradient, while\nterm \"pure drift\"\nwill denote the case\nwith no\ntemperature\ngradient.\n\nThe measured time-spectra are thus qualitatively consistent with the muon\ntrajectories simulation of Fig. 46 (left). We can surely claim that the muons\nmoved from left to right towards the tip of the target, but we cannot im-\nmediately prove that the size of the muon swarm was actually reduced in\nthe y-direction. To demonstrate the actual compression, we need to compare\nthe time-spectra obtained when the temperature gradient is applied with the\ntime spectra obtained with no temperature gradient, i.e, when there is only\ndrift, but no compression.\n\n“Pure drift” If we remove the temperature gradient and set the pressure\nto 3.5 mbar and temperature to 5 K on average3 in the whole target4, the\nmuons simply drift in +x-direction, without any compression in y-direction\n\n3 Note that the actual temperature in the experiment was not completely homogeneous in the\ntarget due to some small thermal radiation: the bottom sapphire plate was at 4 K, while the\ntop one was at 6 K.\n\n4 This is corresponds to the density at y = 0 when we have a temperature gradient from 6 K\nto 18.6 K at 8.6 mbar.\n\n\n\n98 transverse compression\n\n(\"pure drift\"), as shown in the Fig. 46 (right). The obtained time-spectra are\nplotted in Fig. 48 (black points and lines). If we compare these time-spectra\nto the ones with the temperature gradient (Fig. 48, red curves) we see some\ndifferences, but only approximately up to a multiplication factor. Indeed if\nwe scale each of the black time-spectra to have the same number of counts\nas the corresponding red time-spectra, we obtain the dashed black lines in\nFig. 48\n\n5. These scaled time-spectra do not differ anymore significantly from\nthe \"compression\" time-spectra (red lines).\n\nHowever, this finding does not necessarily mean that we have not achieved\nany muon compression in the case with temperature gradient. The problem\nis that the muons that are stopped close to y = 0 drift straight towards\nthe tip, both in the case with and without temperature gradient. This re-\nduces our sensitivity to two different scenarios. One way to overcome thisThe way we select\n\nthe hole is by\nmoving the magnet\n\nsligthly up or\ndown. Since\n\nmuons (at “large”\nmomentum,\n\n∼ 12 MeV/c) follow\nthe magnetic field\nlines, by changing\nthe position of the\nmagnet relative to\nthe position of the\n\ncopper aperture, we\ncan center the\n\nbeam at either of\nthe two holes.\n\nproblem is to modify the copper aperture in front of the target by making\ntwo smaller apertures at y = 4.5 mm (\"top hole\") and y = −4.5 mm (\"bot-\ntom hole\"), instead of one big aperture centered at y = 0 mm, as shown\nin Fig. 45. By injecting the muons through either of two apertures we can\ntarget only a narrow vertical region of the gas, thus creating very distinct\ndensity conditions in \"compression\" and \"pure drift\" measurements. Con-\nsequently, the muon drift angles differ significantly between \"compression\"\nand \"pure drift\" measurements, thereby increasing our sensitivity to distin-\nguishing those two cases.\n\n“top hole” First, the measurements with the “top hole” will be pre-\nsented. The electric field, the gas pressure and the temperature were the\nsame as in the measurements with the big aperture: 6.0-18.6 K at 8.6 mbar\nfor \"compression\" and 5 K at 3.5 mbar for \"pure drift\", with 5.0 kV applied\nin both cases.\n\nGeant4 simulations of the muon trajectories when muons are injected\nthrough the “top hole” for both \"compression\" and \"pure drift\" cases are\nshown in Fig. 49. The main difference we can observe between these two\ncases is that for \"pure drift\" the muons start crashing into the top target wall\nsooner (as early as at 1000 ns).\n\nThe corresponding measured time-spectra of the detectors A1, A2R, A2L\nand A3 are presented in Fig. 51. In the \"compression\" time-spectra (red\npoints), the number of counts in the detectors A3, A2L and A2R first in-\ncreases, then decreases with the time, which suggests that muons \"flew-by\"\nall these three detectors. The number of counts in the time-spectrum of\nthe detector A1 simply increases with time, as before, which tells us that\nthe muons drifted towards the tip of the target. From the position of the\nmaximum number of counts in the time-spectra we can deduce that muons\npassed the vicinity of detector the A3 around 600 ns, then the detector A2LThe region in\n\nproximity of the\ndetector\n\ncorresponds to the\nregion with the\n\nhighest detection\nefficiency\n\n(“acceptance\nregion”).\n\naround 1100 ns, the detector A2R at 1600 ns and finally reached the accep-\n\n5 Such rescaling is allowed since we do not know the absolute stopping efficiencies as the initial\nbeam momentum is not sufficiently well known. The stopping distribution and efficiency\ncould be different between \"compression\" and \"pure drift\" measurements because of their\nvery different He density distributions.\n\n\n\n4.2 results of the 2015 beam-time 99\n\ntance region of the detector A1 around 2000 ns. These numbers fit well with\nthe simulated trajectories of Fig. 49 (left).\n\nThe shape of the \"pure drift\" time-spectra (black points) in this case (using\nthe “top hole”) differs significantly from the \"compression\" time-spectra. In\nall the \"pure drift\" time-spectra the number of counts increases with time\nand after a certain point it stays constant, which means that muons did not\nmanage to \"fly-by\" the detectors, but they crashed into the walls of the target\nbeforehand. The number of counts in the detector A1 stays very low at all\ntimes, suggesting that the muons never reached the tip of the target. This is\nconsistent with the simulated trajectories of Fig. 49 (right).\n\nFigure 49: Projections of the muon trajectories in the yx-plane for the “top hole”\nbeam setting and for the “compression” (left) and “pure drift” density\nconditions (right). The trajectories were simulated in Geant4 assuming\nthe electric field of Fig. 42 (HV = 5.0 kV) and 5 tesla magnetic field.\nThe density distributions in both cases were simulated in Comsol as well\nand imported into the Geant4 simulation. The orange circles indicate the\naperture (“top hole”) position.\n\nTo compare the measurements with the Geant4 simulations more precisely,\nthe simulated muon trajectories of Fig. 49 were convoluted with the simu-\nlated detector acceptance of Fig. 44 to obtain the time-spectra of Fig. 51 (red\nand black lines, for \"compression\" and \"pure drift\", respectively). These\nsimulated time-spectra are then fitted to the measurements using two fit\nparameters:\n\n1. Normalization: to account for the uncertainties in the detection and\nstopping efficiencies,\n\n2. Flat background: to account for a possible misalignment between the\ntarget and the magnetic field axes.\n\nThe fits were performed independently for each detector and each measure-\nment. The global6 reduced chi-square7 is 4.44 for “compression”, and 11.52\n\n6 For all four detectors: A1, A2R, A2L and A3.\n7 In this thesis we are using the following definition of the reduced chi-square:\n\nχ2/ndf =\n1\n\nndf\n\nN∑\ni=1\n\n(ndata,i −nsim,i)\n2\n\nσ2data,i\n, (204)\n\n\n\n100 transverse compression\n\nFigure 50: The projections of the muon trajectories in the yx-plane for the “top\nhole” beam setting and for the “compression” (top left) and “pure drift”\n(top right) density conditions, assuming the target axis is pointing in\nthe (0.018, 0.007, 0.9998) direction (instead of the design value of (0, 0, 1)).\nSuch a misalignment of the target axis with respect to the magnetic field\naxis (in the (0, 0, 1) direction) cause the muon initial position inside the\ntransverse compression target to shift on average by −2 mm in x and\n+3mm in y-direction (for \"compression\") and −1.9mm in x and +1.8mm\nin y-direction (for \"pure drift\"), compared to the positions for no target\nmisalignment (see Fig. 49). The orange circles indicate the aperture (“top\nhole”) position.\n(Bottom left) Zoom of the simulated muon trajectories to show their peri-\nodic pattern when moving close to the top wall. For better visibility, two\ntrajectories are indicated in black. This effect is caused by the electrode\nstructure that is used to define the electric field in the target (bottom\nright). Between two electrodes the electric field is stronger than in the\nrest of the target, while close to the surface of the electrode the elec-\ntric field is relatively weak (because the whole electrode is at the same\npotential). This leads to a pattern in the electric field strength, that in-\nfluences the muon trajectories. When the electric field is relatively weak,\nthe muons follow the electric field direction, causing the muons to crash\ninto the top wall of the target (more details in Section 4.3).\n\nfor “pure drift”, for 172 degrees of freedom8. Such a large reduced chi-\n\nwhere N is the number of bins in the histogram, ndata,i and nsim,i are the numbers of counts\nin the i-th bin of the data and simulation histograms, respectively, σdata,i is the statistical\nuncertainty of ndata,i and ndf is the number of degrees of freedom (equal toN - the number of\nfitted parameters). The statistical uncertainty of the simulated points is significantly smaller\nthan σdata,i, and can therefore be neglected.\n\n8 Note that in order to improve this fit the detector and collimator positions were slightly\nadjusted in the Geant4 simulation (up to 1 mm shift) compared to the design value. Such\nadjustment is justified because the cooling-down process from room temperature to 4 K\nintroduces fairly large uncertainties in the exact positions of the different parts of the setup,\n\n\n\n4.2 results of the 2015 beam-time 101\n\nsquare points to a systematic discrepancy between the simulation and the\nmeasurements. For example, we can see that simulated \"compression\" time-\nspectra are shifted to the right (to later time) compared to the measurements.\n\nThis discrepancy can possibly be explained by the misalignment (tilt) be-\ntween the target axis and the magnetic field axis, which likely develops dur-\ning the setup cool-down. Such misalignment would lead to a shift of the\ninitial position of the beam, and as a result, the times when the muons reach\nvarious detectors would be modified. The \"pure drift\" measurement is es-\npecially sensitive to such misalignment: moving the initial beam position\nchanges the time and position at which muons crash into the wall signifi-\ncantly. In the \"compression\" measurement, such misalignment is less prob-\nlematic, because the temperature gradient ensures that muons reach the tip\nof the target, regardless of their initial position. For this reason, the reduced\nchi-square is typically larger for the “pure drift” measurements compared\nto the “compression” measurements.\n\nThe effects of such misalignment have been investigated by running Geant4\nsimulations for various values and directions of the tilt between the target\nand the magnetic field axes. For each tilt, the simulation was fitted to the\ndata. The best fit for both \"compression\" and \"drift\" simultaneously was ob-\ntained by introducing a tilt (rotation) of −1.10° around the (2.5,−1.0, 0) axis,\nthereby transforming the target axis from (0, 0, 1) to (0.018, 0.007, 0.9998) di-\nrection. This tilt causes the beam to shift on average by −2 mm in x and\n+3 mm in y-direction (for \"compression\") and −1.9 mm in x and +1.8 mm\nin y-direction (for \"pure drift\")9 with respect to the target center. The new\nsimulated muon trajectories are shown in Fig. 50, while the simulated time-\nspectra are presented in Fig. 52.\n\nFitting the simulated time-spectra that include the above described tilt to\nthe measured time-spectra improves the global reduced chi-square from 4.44\n\nto 2.92 for \"compression\" and from 11.52 to 2.56 for “pure drift” (with 172\n\ndegrees of freedom in each case). Further improvements of the fit would\ncertainly be possible by running the simulation in even smaller steps of the\ntilt angles, tilt directions and detector positions. However, in the presence of\nsuch a tilt, the measurement becomes very sensitive to the muon beam mo-\nmentum distribution. The reason is that the beam momentum determines\nthe muon stopping position in the z-direction. Muons of higher momentum\nare stopped further downstream compared to the muons of lower momen-\ntum, which leads in comparison to a larger offset of the muon initial posi-\ntion from the design position(for the same target axis tilt). Unfortunately,\nthe exact muon beam momentum distribution is not sufficiently well known.\nTherefore, proper treatment of the misalignment would require finely tun-\ning the tilt along with the momentum distribution and the detector positions,\nwhich would be computationally very expensive.\n\ndue to different thermal contraction of various materials used. Additionally, the scintillator\npositions are known with only 0.5 mm uncertainty due to mechanics and Teflon scintillator\nwrapping tolerance.\n\n9 Note that the shift of the initial beam position is different for \"compression\" and \"pure drift\"\nmeasurements due to different gas densities in the target, that lead to different muon stop\ndistributions.\n\n\n\n102 transverse compression\n\nFigure 51: Measured time-spectra of the detectors A1, A2R, A2L and A3 for the “top\nhole” beam setting and for the “compression” (red points) and “pure\ndrift” (black points) conditions. In both cases the maximum applied HV\nis 5 kV. The simulated time-spectra (lines), corresponding to the Geant4\nsimulations of Fig. 49, were fitted to the measured time-spectra using\ntwo fit parameters (normalization and flat background). The fits were\nperformed independently for each measurement and for each detector.\n\nThe simulations presented in this chapter already reproduce well the main\nfeatures of the measured time spectra, such as times at which the time-\nspectrum reaches the maximum and the time when the time-spectrum be-\ncomes flat. Hence, we can conclude that the measurements demonstrate\nconvincingly the transverse compression of the muon beam.\n\n“bottom hole” For completeness, the measurements with muons in-\njected through the “bottom hole” will be presented here.\n\nCorresponding Geant4 simulations of muon trajectories are shown in Fig. 53,\nfor both \"compression\" (left) and \"pure drift\" (right). The measured time-\nspectra, along with the fitted simulated time-spectra are presented in Fig. 54\n\nfor the detectors A1, A2R, A2L and A3. The shape of the measured \"com-\npression\" time-spectra (red points) is qualitatively similar to the “top hole”\n\"compression\" time-spectra, with characteristic rise and subsequent fall in\nthe number of counts with the time in the detectors A3 and A2L and simple\nrise of the number of counts with time in the detectors A2R and A1. The\nmain difference is that the muon drift in this case is slower, as indicated by\nthe times when muons pass the various detectors and reach the tip of the\ntarget, which are up to 1500 ns slower compared to the “top hole” measure-\n\n\n\n4.2 results of the 2015 beam-time 103\n\nFigure 52: Measured time-spectra of the detectors A1, A2R, A2L and A3 for the “top\nhole” beam setting and for the “compression” (red points) and “pure\ndrift” (black points) conditions. The simulated time-spectra (lines) as-\nsuming the small target tilt (described in the text), corresponding to the\nGeant4 simulations of Fig. 50, were fitted to the measured time-spectra\nusing two fit parameters (normalization and flat background). The fits\nwere performed independently for each measurement and for each detec-\ntor. Introduction of the target misalignment with respect to the magnetic\nfield axis improves the fit significantly (compare with the fit of Fig. 51).\n\nments. Indeed, when muons are injected through the “bottom hole”, they\ndrift through a region of higher gas density compared to the “top hole” in-\njection, which makes their drift slower. This behavior is well visible when\ncomparing the time evolution of the muon trajectories in Fig. 53 (“bottom\nhole” injection) with the time evolution of Fig. 49 (“top hole” injection).\n\nFitting the simulated \"compression\" time-spectra to the measurements of\nFig. 54 with two free parameters, normalization and background, as before,\ngives us the reduced chi-square of 1.66 for 300 degrees of freedom.\n\nIn contrast to the “top-hole” measurements, the \"pure drift\" measured\ntime-spectra (black points) are almost identical to the \"compression\" time-\nspectra (red points). One reason is certainly related the detector resolu-\ntion, which is worse for the “bottom-hole” measurements because muons\nare drifting at larger distance from the detectors. However, even with poorer\ndetector resolution we would expect a clearer contrast between \"compres-\nsion\" and \"pure drift\" measurements. Consider the simulated time-spectra\nfor \"compression\" (red lines) and \"pure drift\" (black lines) in Fig. 54 : the\ndistinction between these two cases is clearly better than observed in the\nmeasurements. This discrepancy between simulations and measurements\n\n\n\n104 transverse compression\n\nFigure 53: Projections of the muon trajectories in the yx-plane for the “bottom hole”\nbeam setting and for “compression” (left) and “pure drift” (right) density\nconditions. The trajectories were simulated in Geant4 assuming the elec-\ntric field of Fig. 42 (with maximum HV of 5.0 kV) and a 5 tesla magnetic\nfield. The density distributions in both cases were simulated in Com-\nsol as well and imported into the Geant4 simulation. The orange circles\nindicate the aperture (“bottom hole”) position.\n\nFigure 54: Measured time-spectra of the detectors A1, A2R, A2L and A3 for the\n“bottom hole” beam setting and for the “compression” (red points) and\n“pure drift” (black points) conditions. In both cases the maximum ap-\nplied HV was 5 kV. The simulated time-spectra (lines), corresponding\nto the Geant4 simulations of Fig. 53, were fitted to the measured time-\nspectra using two fit parameters (normalization and flat background).\nThe fits were performed independently for each measurement and for\neach detector.\n\n\n\n4.3 electric field scan 105\n\nalso manifests itself in the goodness of the fit for the \"pure drift\" case, where\nwe obtain a reduced chi-square of 20.3 for 300 degrees of freedom.\n\nFollowing the same reasoning as in the previous section, we introduce a\nsmall tilt of the target axis with respect to the magnetic field axis of −1.10°\naround the (0.25,−1.0, 0) axis, thereby shifting the target axis from initial\n(0, 0, 1) to (0.019, 0.005, 0.9998) direction. This tilt shifts the initial muon stop-\nping distribution on average by −3.5 mm in x and +0.2 mm in y-direction\n(for \"compression\") and −3.7 mm in x and +0.25 mm in y-direction (for\n\"pure drift\"). It is justified to use a different value of the tilt than for the\n“top-hole” measurements, because to inject the muons through the “bottom\nhole” we needed to adjust the magnet position and orientation.\n\nFigure 55: Projections of the muon trajectories in the yx-plane for the “bottom hole”\nbeam setting and for the “compression” (left) and “pure drift” (right) den-\nsity conditions, and the target axis pointing in the (0.019, 0.005, 0.9998)\ndirection (instead of the design value of (0, 0, 1)). Such a misalignment\nof the target axis with respect to the magnetic field axis cause the muon\ninitial position inside the transverse compression target to shift on aver-\nage by 3.5 mm in x and +0.2 mm in y-direction (for \"compression\") and\n−3.7mm in x and +0.25mm in y-direction (for \"pure drift\"), compared to\nthe positions for no target misalignment (see Fig. 53). The orange circles\nindicate the aperture (“bottom hole”) position.\n\nIntroducing this kind of tilt improves the reduced chi-square from 1.66 to\n1.40 for \"compression\" and from 20.31 to 2.68 for \"pure drift\", in both cases\nfor 300 degrees of freedom, as shown in Fig. 56. Again, it would be possible\nto further improve the fit by tuning various simulation parameters more\nfinely. However, little further insight would be gained from this optimization\ngiven the limited sensitivity of the measurement to the difference between\n\"compression\" and \"pure drift\".\n\n4.3 electric field scan\n\nThus far, we have investigated the effect of the vertical gas density gradient\non the muon drift. The section that follows moves on to consider the effect\nof the electric field strength (at the same density conditions) on the muon\nmotion, allowing us to examine the properties of the muon drift as a function\nof its energy and to validate the Geant4 simulations.\n\n\n\n106 transverse compression\n\nFigure 56: Measured time-spectra of the detectors A1, A2R, A2L and A3 for the “bot-\ntom hole” beam setting and under the “compression” (red points) and\n“pure drift” (black points) conditions. The simulated time-spectra (lines)\nassuming the small target tilt, corresponding to the Geant4 simulations of\nFig. 55, were fitted to the measured time-spectra using two fit parameters\n(normalization and flat background). The fits were performed indepen-\ndently for each measurement and for each detector. Introduction of the\ntarget misalignment with respect to the magnetic field axis improves the\nfit significantly (compare with the fit of Fig. 54).\n\nAs mentioned already in Chapter 1 and Chapter 3, the drift angle ϑ of the\nmuon relative to the Ê× B̂ direction is proportional to the collision frequency\nνc of the muons with He atoms:\n\ntan ϑ =\nνc\n\nω\n, (205)\n\nwith ω being the muon cyclotron frequency. The collision frequency νc is\ndefined as\n\nνc = NσMT (Eµ) |vr| , (206)\n\nwhereN is the helium gas density (in cm−3), σMT (Eµ) is the energy-dependent\nmuon-He momentum transfer cross section and vr is muon-He relative ve-\nlocity.\n\nFor low muon energies, below . 1 eV, σMT is proportional to 1/\n√\nECM ∝\n\n1/ |vr| [18] so that the collision frequency becomes independent of the muon\nenergy. For energies larger than 1 eV, the product σMT |vr| decreases with\nenergy, as shown in Fig. 57, and, as a result, the collision frequency and the\nmuon drift angle ϑ decrease with increasing muon energy. Since the muon\n\n\n\n4.3 electric field scan 107\n\nFigure 57: Plot of the product σMT |vr| as a function of muon energy in the labora-\ntory frame.\n\nenergy in the He gas depends on the value of the electric field applied10, the\nmuon drift direction will likewise be affected by the electric field strength.\n\nThis consideration is important for the design of the transverse compres-\nsion target. If the electric field is too weak, the collision frequency will be\nlarge and the muon will simply drift along the electric field direction at all\ngas densities present in the target. Thus, there will be no (or very inefficient)\ntransverse compression and the muons will crash into the top wall of the\ntarget.\n\nThe dependence of the muon motion on the electric field strength was\nstudied for both “top” and “bottom hole” muon beam injections, and for\n\"compression\" and \"pure drift\" density conditions. To avoid repetition, only\nmeasurements and simulations for the “top hole” will be described in more\ndetail.\n\n“compression”, “top hole” First, the measurements and simulations\nfor the “top hole” injection at 8.6 mbar with temperature gradient from 6 K\nto 18.6 K are presented (\"compression\" conditions).\n\nThe simulated muon trajectories for various electric field strengths (max-\nimum applied HV) are shown in Fig. 58. Note that the simulations already\naccount for the tilt between target and magnetic field axis: the target was as-\nsumed to point in (0.018, 0.007, 0.9998) direction, determined by minimizing\nthe chi-square for the 5.0 kV measurements (see previous section). We can\nsee that, with increasing electric field strength, the muon drift direction ap-\nproaches the Ê× B̂-direction. This is consistent with the explanation above:\n\n10 The kinetic energy of the muon increases with increasing electric field strength.\n\n\n\n108 transverse compression\n\nwith increasing electric field strength, the muon energy increases and thus\nthe collision frequency decreases, leading to smaller angle ϑ with respect to\nthe Ê× B̂-direction.\n\nThe measured time-spectra, along with the fitted simulated time-spectra\nare shown in Fig. 59. The measurements were normalized to have the same\ntotal number of counts in the entrance detector. At smaller applied HV we\ncan observe a slower drift, as can be deduced from the broadening of the\nfly-by peak in the time-spectra of the detectors A2L and A3. The maximum\nnumber of counts in A1 decreases with decreasing HV, which means that\nless and less muons reach the tip of the target. This is confirmed by the\ndetector A2R as well - for lower HV, the muons do not manage to pass by\nthis detector, but they crash into the wall before. This is consistent with the\nGeant4 simulation of the trajectories of Fig. 58.\n\nThe simulations for each value of the high voltage and for each detector\nwere fitted independently with two fit parameters: normalization and back-\nground. The obtained reduced chi-square values range from 1.81 to 2.92 (for\n172 degrees of freedom in all the cases), as summarized in Table 3.\n\nHV (kV) 3.00 4.00 4.50 4.75 5.00 5.25 5.50 5.75\n\n“top hole” “compression” − − 2.57 2.90 2.92 2.84 1.81 2.56\n\n“top hole” “pure drift” 1.95 2.35 3.34 3.14 2.56 2.11 − −\n\n“bottom hole” “compression” − − 3.69 1.40 1.40 2.14 2.70 -\n\n“bottom hole” “pure drift” 2.84 9.96 11.66 7.00 2.68 8.88 − −\n\nTable 3: Reduced chi-square for various measurements (“top hole”, “bottom hole”,\n“compression”, “pure drift”) and for various applied HV. In all cases the\nnumber of degrees of freedom is 172. The symbol − signifies that no mea-\nsurement was done at this value of the HV.\n\nIt was not possible to perform a simultaneous fit of the time-spectra for\nvarious HV values. We indeed observed that the normalization giving the\nbest fit depends on the HV (see Fig. 60). Such a behavior could indicate\na mismatch of the initial muon stop distribution between the simulations\nand experiment. Indeed, the exact muon stopping distribution is hardly\nreproducible as it depends on several parameters that are not well known:\ndistribution of the muon beam transverse and longitudinal momenta, the\nenergy losses in various foils, the target misalignment. Also, a proper treat-\nment of the misalignment would require optimizing the target tilt for all the\nHV values simultaneously (and not just for 5 kV).\n\nEven though the fits are not perfect, the simulation still reproduces the\nmain characteristics of the muon motion correctly: mainly the times at which\nmuons pass through the various detector acceptance regions are roughly con-\nsistent between the simulation and measurement. This is visible from com-\nparing the times when time-spectra reach the maximum number of counts\nand/or when they become flat. These times are dependent on the muon\ndrift velocity, which is not influenced strongly by the various misalignments.\n\n\n\n4.3 electric field scan 109\n\nFigure 58: Projections of the muon trajectories in the yx-plane for the “top hole”\nbeam setting and for the “compression” density conditions (6.0− 18.6 K\nat 8.6 mbar), for various applied HV. The target axis was assumed to\npoint in the (0.018, 0.007, 0.9998) direction (instead of the design value of\n(0, 0, 1)).\n\nFigure 59: Measured time-spectra of the detectors A1, A2R, A2L and A3 for the\n“top hole” beam setting and under the “compression” density conditions\nand for various applied HV. The simulated time-spectra (lines) assuming\nsmall target tilt, corresponding to the Geant4 simulations of Fig. 58, were\nfitted to the measured time-spectra using two fit parameters (normaliza-\ntion and flat background). The fits were performed independently for\neach measurement and for each detector.\n\n\n\n110 transverse compression\n\nFigure 60: Dependence of the normalization fit parameter on the applied HV, for\nvarious detectors, corresponding to the fits of Fig. 59 (“top hole”, “com-\npression”).\n\nContrarily, the exact shape of the time-spectra depends strongly on the initial\nmuon stopping position.\n\n“pure drift”, “top hole” The muon trajectories simulated assuming\nno density gradient, i. e., for \"pure drift\" conditions, are shown in Fig. 61.\nAgain, the target was oriented along (0.018, 0.007, 0.9998) direction in the\nsimulation. We can observe a similar trend as before: with increasing elec-\ntric field strength, the Ê× B̂ component of the muon drift velocity increases.\nThe absence of a temperature gradient makes this behavior even more pro-\nnounced compared to the “top hole” \"compression\" measurements.\n\nThe measured and simulated time-spectra for the “pure drift” are shown\nin Fig. 62. Most of the time-spectra exhibit similar shape: increase of counts\nwith time, then flattening. The stronger the electric field, the larger is the\nmaximum number of counts, meaning more muons reached the acceptance\nregion of the detectors. This is consistent with the muon trajectories of\nFig. 61 - with increasing electric field, the muon drift direction approaches\nthe Ê× B̂ direction, so that muons can avoid crashing into the top target wall\nfor a longer time. For the detector A3 at HV=5.0 kV and HV=5.25 kV, the\nusual fly-by signature is visible: increase then decrease of counts with the\ntime, which also supports the Geant4 trajectory simulations.\n\nFitting the simulated time-spectra to the measurements for each HV and\neach detector separately produces the reduced chi-square values from 1.95\n\nto 3.34, for 172 degrees of freedom, as shown in the Table 3.\n\n“compression”, “bottom hole” The simulated muon trajectories at\nvarious electric field strengths are shown in Fig. 63. The target axis was\nassumed to point in the (0.019, 0.005, 0.9998) direction, determined by opti-\nmizing the fit of the 5.0 kV measurements (see previous section).\n\n\n\n4.3 electric field scan 111\n\nFigure 61: Projections of the muon trajectories in the yx-plane for the “top hole”\nbeam setting and for the “pure drift” density conditions (5 K on average\nat 3.5 mbar), for various applied HV. The target axis was assumed to\npoint in the (0.018, 0.007, 0.9998) direction (instead of the design value of\n(0, 0, 1)).\n\nFigure 62: Measured time-spectra of the detectors A1, A2R, A2L and A3 for the\n“top hole” beam setting and for the “pure drift” density conditions and\nfor various applied HV. The simulated time-spectra (lines) assuming the\nsmall target tilt, corresponding to the Geant4 simulations of Fig. 61, were\nfitted to the measured time-spectra using two fit parameters (normaliza-\ntion and flat background). The fits were performed independently for\neach measurement and for each detector.\n\n\n\n112 transverse compression\n\nThe measured time-spectra, along with the fitted simulated time-spectra\nare shown in Fig. 64. Again, nice progression is visible in the time-spectra\nas we increase the high voltage: muons reach the detector acceptance re-\ngion faster and we get more pronounced signals (shape, maximum number\nof counts) in the detectors. The difference between the various HV mea-\nsurements is somewhat less distinct than in the “top hole” measurements\ndue to muons drifting at larger distances from the detectors. The reduced\nchi-square values are summarized in Table 3.\n\n“pure drift”, “bottom hole” Injecting the muons through the “bot-\ntom hole” into the target with no density gradient and changing only the\nstrength of the applied electric field results in the muon trajectories of Fig. 65.\n\nThe corresponding measured and simulated time-spectra are plotted in\nFig. 66. Fitting the simulated time-spectra to the measurements gives us\nreduced chi-squares from 2.68 to 11.68 for 172 degrees of freedom (see Ta-\nble 3). Systematic discrepancies between measurements and simulation are\nespecially prominent at these conditions: the value of the fitted normaliza-\ntion for all four detectors depends strongly on the value of the HV, as visible\nin Fig. 67. Such a discrepancy is not surprising as the measurements at\nthese conditions (“bottom hole”, “pure drift”) are particularly sensitive to\nthe initial muon beam position and momentum distribution.\n\n\n\n4.3 electric field scan 113\n\nFigure 63: Projections of the muon trajectories in the yx-plane for the “bottom hole”\nbeam setting and for the “compression” density conditions (6.0− 18.6 K\nat 8.6 mbar), for various applied HV. The target axis was assumed to\npoint in the (0.019, 0.005, 0.9998) direction (instead of the design value of\n(0, 0, 1)).\n\nFigure 64: Measured time-spectra of the detectors A1, A2R, A2L and A3 for the\n“bottom hole” beam setting and for the “compression” density conditions\nand for various applied HV. The simulated time-spectra (lines) assuming\nsmall target tilt, corresponding to the Geant4 simulations of Fig. 63, were\nfitted to the measured time-spectra using two fit parameters (normaliza-\ntion and flat background). The fits were performed independently for\neach measurement and for each detector.\n\n\n\n114 transverse compression\n\nFigure 65: Projections of the muon trajectories in the yx-plane for the “bottom hole”\nbeam setting and for the “pure drift” density conditions (5 K on average\nat 3.5 mbar), for various applied HV. The target axis was assumed to\npoint in the (0.019, 0.005, 0.9998) direction (instead of the design value of\n(0, 0, 1)).\n\nFigure 66: Measured time-spectra of the detectors A1, A2R, A2L and A3 for the “bot-\ntom hole” beam setting and under the “pure drift” density conditions\nand for various applied HV. The simulated time-spectra (lines) assuming\nthe small target tilt, corresponding to the Geant4 simulations of Fig. 65,\nwere fitted to the measured time-spectra using two fit parameters (nor-\nmalization and flat background). The fits were performed independently\nfor each measurement and for each detector.\n\n\n\n4.4 conclusions 115\n\nFigure 67: Dependence of the normalization fit parameter on the applied HV, for\nthe fits of Fig. 66 (“bottom hole”, “pure drift”).\n\n4.4 conclusions\n\nThis chapter presented the results of the transverse compression stage of\nthe 2015 beam-time. The challenging cryogenic setup required for this mea-\nsurement was realized successfully, allowing us to observe the muon beam\ntransverse compression for the first time. One critical aspect of this demon-\nstration was distinguishing between a simple muon drift versus drift with\nsimultaneous reduction of the beam transverse size. Such distinction was\naccomplished by performing the measurements of the muon motion with\nand without vertical temperature gradient, which is needed for the trans-\nverse compression, and by injecting the muon beam separately into different\ndensity regions inside the transverse compression target.\n\nThe muon motion corresponding to the realistic experimental conditions\nwas simulated using the Geant4 simulation described in Chapter 3. Very\ngood agreement between the simulations and measurements could be achieved\nafter accounting for small target and detector misalignments.\n\nBesides demonstrating the transverse compression, the dependence of the\nmuon drift on the applied electric field strength was explored experimen-\ntally. A comparison with Geant4 simulations was also made for these mea-\nsurements. Relatively good agreement could be observed in most cases (at\nvarious electric field strengths). One of the most important features of the\nmuon motion (drift velocity) is consistent between the corresponding simu-\nlations and measurements. The drift velocity is determined by the energy-\ndependent muon-He elastic collision cross section, allowing us to validate\nour modeling of the low-energy muon-He elastic collisions, as described in\nChapter 3.\n\nStill, some electric-field-dependent systematic discrepancies of the simula-\ntion fits to the measurements have been observed. They are most likely re-\n\n\n\n116 transverse compression\n\nlated to the uncertainties of the muon beam initial position and momentum\ndistribution. This is supported by the fact that the discrepancies are larger\nfor the measurements with no density gradient applied, which are more sen-\nsitive to the initial beam parameters compared to the measurements with the\ndensity gradient applied. These discrepancies could also be partially caused\nby the lack of precise definition of the electric field in the tip of the target. In-\ndeed, typically, the various measurements with the same density and beam\nconditions are well described by one common set of fit parameters at low\nelectric field strengths, while at higher field strengths the fit parameters take\ndifferent values. The important difference between low and high electric\nfield measurements is that in the former case, the muons never reach the tip\nof the target, so that the measured time-spectra are not sensitive to the exact\nelectric field in tip of the target.\n\nNevertheless, we can conclude that the muon motion is sufficiently well\ndescribed by the existing simulation, within the limitations of the experi-\nmental setup implemented for these measurements. Further improvements\nof the measurements could be made by better beam and target alignment.\nOne interesting area of future studies could be the investigation of the sen-\nsitivity of the measurements to the small variations of the elastic collision\ncross sections implemented in the Geant4 simulation. This could be done\nwith the already obtained data.\n\n\n\n5\nL O N G I T U D I N A L C O M P R E S S I O N\n\nThe longitudinal compression stage is the second stage of the muCool beam-\nline. A preliminary test of this stage was performed for the first time in\n2011 [19]. Although the longitudinal compression of the muon beam was\ndemonstrated, it was not possible to quantify precisely the compression effi-\nciency due to impurities present in the helium gas and due to a large muon\nbeam related background. Therefore, an improved test of the longitudinal\ncompression stage was performed in 2014, which allowed us to extract the\ncompression efficiency for the longitudinal compression stage and to mea-\nsure the muon z-distribution during the compression.\n\nAdditionally, in 2015 the longitudinal compression stage was modified to\ninclude the vertical (in the y-direction) electric field component, which is\nneeded to transport the muons through various stages (in the x-direction).\nThe resultant Ê× B̂ drift superimposed on the longitudinal compression was\nmeasured for the first time.\n\nIn this chapter the results from the 2014 and 2015 beam-times are pre-\nsented, along with the corresponding simulations and analysis. The text of\nthis chapter is in large part reproduced from the author’s publication [20].\n\n5.1 improved test of the longitudinal compression\n\nIn order to test the longitudinal compression, a muon beam with 11 MeV/c\nmomentum was injected longitudinally into a target containing He gas of\nfew mbar pressure at room temperature. The measurements were performed\nat the πE1 beam-line at Paul Scherrer Institute, providing about 104 µ+/s\nat 11 MeV/c. The setup used for this measurement is sketched in Fig. 68.\nThe muons first have to pass through a 55 µm thick entrance detector (D0),\ngiving the initial time t0 = 0, then through a 2 µm Mylar target window\nenclosing the He gas. Only a small1 fraction O(1%) of the muons producing\na signal in the D0 detector stop in the He gas, while the remaining muons\nstop either in the entrance detector or target window (region I in Fig. 68) or\nin the downstream wall of the target (region II in Fig. 68).\n\nelectric fields The longitudinal compression target had a transverse\ncross section of 12 × 12 mm2, and an “active\" length (in the z-direction),\nwhere the electric field was defined, of about 200 mm, see Fig. 69. The side\nwalls of the target were lined with gold electrodes that created a V-shaped\nelectric potential with minimum in the center of the target cell at z = 0. Such\n\n1 The stopping efficiency is low in this case because of the low gas density in the longitudinal\ncompression target (room temperature target). In the final setup the beam will be stopped in\na cryogenic gas (in the transverse compression stage), so that the majority of the high-energy\nmuon beam is stopped in the gas.\n\n117\n\n\n\n118 longitudinal compression\n\nFigure 68: Sketch of the He gas target used to measure the longitudinal compression\n(not to scale). The muon beam passes through an entrance detector D0,\ntarget window (region I) and enters the He gas (indicated in pink). Some\nof the muons are stopped in the gas and subjected to the electric field,\npointing towards the center of the target. A large fraction of the muons\npass through the target and stop in region II. Several scintillators (S1, . . .,\nS21 , T1, T2) detect the positrons from muons decaying in their vicinity.\n\na potential is achieved by grounding the two most external electrodes and\nby applying a negative HV to the electrodes in the center (at z = 0). All\nthe other electrodes are connected to these via SMD resistors, acting as a\nvoltage divider, so that the potential on the electrodes drops linearly from\n0 V to -HV when moving from the upstream and downstream ends of the\ntarget towards the center of the target. This choice of the electric potential\nproduces an electric field pointing towards z = 0, which caused the muons\nto move along the ±z-direction towards the potential minimum. A picture of\nthe realized target and the corresponding potential distribution at the target\nwalls are shown in Figs. 69 and 70, respectively. The \"cross\" geometry of\nthe target allowed to keep all the soldering and the voltage divider (SMD\nresistors) outside the gas volume, ensuring the needed gas purity.\n\nTo simulate the electric field inside the longitudinal compression target,\nthe potential distribution on all electrodes is solved for using the \"Electric\ncurrents\" module of COMSOL Multiphysics®, as shown in Fig. 70. This\npotential is then used as a boundary condition in the next step, where the\npotential in the whole target volume is calculated using the \"Electrostatics\"\nmodule. The obtained electric field and potential in the yz-plane at for x = 0\nare shown in Fig. 71.\n\nThe plot of the electric potential along the z-axis is shown in Fig. 72 (green\ncurve). Around z = 0, the slope of the electric potential curve decreases\n(which manifests itself as a rounding of the curve), leading to a reduced\nelectric field in the z−direction around z = 0. Indeed, from the Maxwell\nequation ∇ · E = 0, it follows that the electric field cannot suddenly change\nits direction. For the same reason, the electric field develops an additional ra-\ndial component at z = 0, as shown in the Fig. 73. This component is included\nautomatically in the COMSOL Multiphysics® Multiphysics® simulation but\nit is practically so small that it does not affect significantly the muon motion\n\n\n\n5.1 improved test of the longitudinal compression 119\n\nFigure 69: Photo of the longitudinal compression target used in the 2014 beam-time,\nformed by gluing together four glass plates lined with electrodes..\n\nFigure 70: Electrodes that define the electric field required for the longitudinal com-\npression. The color scale shows the potential distribution on the walls of\nthe longitudinal target, simulated with COMSOL Multiphysics®. Such\npotential distribution produces an electric field of 50 V/cm strength.\n\nduring the compression, except at late times, after the longitudinal compres-\nsion is already finished, as will be shown later.\n\n\n\n120 longitudinal compression\n\nFigure 71: Electric field (black arrows) and electric potential (colored contours) in\nthe yz-plane at x = 0 of the longitudinal target simulated with COM-\nSOL Multiphysics®. In the simulation the potentials on the electrodes of\nFig. 71 were used as boundary condition.\n\nFigure 72: Simulated electric potential along the z-axis (x = y = 0). In the sim-\nulation the potentials on electrodes of Fig. 71 were used as boundary\ncondition.\n\nmuon trajectories simulation Muon trajectories in the electric fields\nof Fig. 71, 5 tesla magnetic field pointing in the −z-direction and in 5mbar of\nHe gas at room temperature were simulated using Geant4 simulation. This\nsimulation included the most relevant muon-He low-energy interactions, as\ndescribed in Chapter 3. At t = 0 the high-energy muon beam of 11.4 MeV/c\nmomentum and 3% momentum spread is generated in front of the entrance\ndetector. After passing through the various foils, a small fraction of muons\nis stopped in the gas target. The z-distribution of the stopped muons (at\nt = 0.15 µs) is shown in Fig. 75 (solid black line). Due to a small transverse\nsize of the target, a fraction of muons is also scattered into the walls of the\ntarget (shaded black area of Fig. 75). Once inside the target, the muons start\nmoving under the influence of the applied electric fields, towards the target\ncenter at z = 0. The muon distribution at t = 2 µs, when the longitudi-\nnal compression is almost completed, is also shown in Fig. 75 (solid green\nline). Note that the number of counts has been scaled with exp(t/τ), where\nτ = 2197 ns is the muon lifetime, in order to compensate for muon decay\nand to better highlight the muon swarm compression.\n\n\n\n5.1 improved test of the longitudinal compression 121\n\nAt late times, after the compression is already finished, the muons that are\nat z = 0 start drifting slowly under the influence of the radial component\nof the electric field in the center of the target (see Fig. 73). Since the radial\ncomponent of the electric field is perpendicular to the magnetic field, the\nmuons drift in the Ê× B̂-direction, giving rise to the slow rotation around\nthe z-axis, as visible in Fig. 74.\n\nFigure 73: Simulated electric field lines (black lines) and potential (color scale) in the\nyx-plane at z = 0 (center of the target). In the simulation the potentials on\nthe electrodes as shown in Fig. 71 were applied. This radial component\nof the electric field in the center of the target causes the slow drift of the\nmuons around the z-axis, as shown in Fig. 73.\n\nUnder such conditions and neglecting muon decay, at t = 2 µs, 63% of\nthe muons that were in the active region at t = 0.15 µs are still within the\nactive region, with 90% of those already within z = ±5 mm. Moreover, 50%\nof all muons in the active region at t = 2 µs are already compressed in the\ncenter within an even smaller region of z = ±1 mm. Therefore, the fraction\nof muons within the region of z = ±1 mm increases by about a factor of 20\n\nbetween 0.15 µs and 2 µs.\nThe other 37% of the muons that were initially in the active region are lost\n\nthrough two main mechanisms: 26% through muonium formation and 11%\nthrough scattering out of the active region due to low-energy elastic colli-\nsions. A muon bound in the neutral muonium atom is not confined by the\n5 tesla magnetic field, causing it to fly into the walls of the target. The losses\ndue to scattering of µ+ are relevant only when the muon beam is injected\ninto the longitudinal target at keV energies, because the µ+ mean free path\nat these energies can be up to a few cm. These losses will therefore be absent\nin the final setup, in which muons enter the longitudinal compression stage\nfrom the transverse compression stage at eV energies. Indeed, at such ener-\n\n\n\n122 longitudinal compression\n\nFigure 74: Muon trajectories in the center of the target (at z = 0) at late times, i. e.,\nwhen the longitudinal compression is already finished. The slow rota-\ntion around the z-axis is caused by the Ê× B̂ drift related to the radial\ncomponent of the electric field shown in Fig. 73.\n\ngies, the µ+ mean free path is sub-mm. Similarly, the losses due to muonium\nformation will be strongly reduced in the final setup - there the muonium is\nmostly formed during the slowing-down of the high-energy muon beam in\nthe cryogenic He gas. Indeed, at these high densities, the Mu mean free path\nis much smaller than in the longitudinal compression target, and thus Mu\natoms are unlikely to fly into the target walls before undergoing a collision\nthat re-ionizes them.\n\nmeasurement of the muon longitudinal compression The muon\nswarm movement was measured experimentally by placing 21 identical plas-\ntic (BC-404) scintillators (S1 to S21 ) along the target z-axis, as shown in\nFigs. 68 and 76. The scintillators had dimensions of 30 × 5 × 6 mm3 and\nwere wrapped in Teflon™ tape. When positrons from muon decays pass\nthrough the scintillators they produce scintillation light which is then read\nout by Silicon Photomultipliers (SiPMs) with 3× 3 mm2 active area. The effi-\nciency of some of these scintillators as a function of muon decay z-position is\nplotted in Fig. 77. The detection efficiency εi(z) of the detector Si is defined\nas:\n\nεi(z) =\nNidet(z)\n\nNgen(z)\n, (207)\n\n\n\n5.1 improved test of the longitudinal compression 123\n\nFigure 75: Simulated muon distribution along the z-axis for various times. The data\nhas been corrected for the finite muon lifetime by multiplying the counts\nwith et/2197ns. As can be seen, for t = 2 µs most muons are close to the\ncenter of the active region. The shaded areas show the fraction of the µ+\n\nstopped in the walls of the target. At t = 2 µs, the distribution of the µ+\n\nadhering to the target walls is not flat anymore. The reason for the peak\naround z = 0 is that some µ+ are scattered into the wall while drifting\ntowards the center of the target.\n\nwhere Ngen(z) is the number of muons generated in Geant4 simulation at\nposition z, and Nidet(z) is the number of positrons detected in the detector\nSi, produced from the muon decay occurring at position z. Note that in\nFig. 77, the detector efficiencies of the various detectors were normalized to\nhave maximum equal to 1, for easier comparison. Each detector (S1-S21) has\nan average geometrical acceptance in the z-direction of 16.5 mm (FWHM).\n\nThe number of measured positrons in each scintillator is presented for\nt = 150 ns (black dots) and t = 2000 ns (green dots) in Fig. 78. Note that the\nnumber of counts has been scaled with et/2197 ns to compensate for muon\ndecay.\n\nIn Fig. 78 the measured µ+ distributions (dots) are compared to the Geant4\nsimulations (curves). The simulated number of positron hits Ni(t) (in a\nsmall time window ∆t, i. e., between t and t+ ∆t) in each of the detectors\nSi is obtained by folding the simulated spatial distribution of the muon\nswarm n(z, t), given in Fig. 75, with the detection efficiency εi(z) of the\ncorresponding detector, given by Eq. (207) and in Fig. 77:\n\nNi(t) =\n∆t\n\nτ\n\n∫\nn(z, t)εi(z)dz, (208)\n\nwhere τ = 2197 ns is the muon lifetime. The factor ∆t/τ in front of the inte-\ngral is needed because the number of muons decaying at a certain position z\n\n\n\n124 longitudinal compression\n\nFigure 76: Photo of the detectors (S1-S21) used for measuring the muon z-\ndistribution (see Fig. 68). The plastic scintillators were mounted inside\nthe brass collimator to improve their spatial resolution.\n\nFigure 77: Simulated geometrical acceptance of the scintillators S1-S21 (only every\nsecond detector is shown) versus the z-position. Shown is the normalized\nprobability that a positron from a µ+ decay is detected in the correspond-\ning scintillator. For comparison, the geometrical acceptance of the T1 &\nT2 coincidence is also plotted (red line). The maximum of the efficiency\nfor each detector was normalized to 1 to highlight the different geomet-\nrical resolutions.\n\nand within a small time window ∆t is approximately given by ∆t\nτ n(z, t) (on\n\naverage).\nThese positron hits originate not only from muons in the active region,\n\nbut also from muons from the regions where the electric field is not well\n\n\n\n5.1 improved test of the longitudinal compression 125\n\nFigure 78: Measured (dots) and simulated (lines) positron hits in the 21 scintillators\nshown in Fig. 68, for t = 150 ns and t = 2000 ns. The center-center\ndistance between two adjacent scintillators is 10mm. The simulation data\nis the same as in Fig. 75, but convoluted with the geometrical acceptance\nof the detectors. The width of the peak at t = 2 µs is due to the large\ngeometrical acceptance of the 21 scintillators and does not reflect the\nwidth of the muon swarm directly (compare with Fig. 75). The shaded\nareas show the simulated contribution to the positron hits from muons\nstopped in regions I and II. The data has been corrected for the finite\nmuon lifetime by multiplying the counts with et/2198ns.\n\ndefined and from the target walls, giving rise to a substantial background.\nThis background is dominated by positron hits from µ+ that stop in the\nregions I and II (as defined in Fig. 68). The background is larger for the\ndetectors placed at the periphery of the active region (S1, S2 and S20 , S21\n), as shown in Fig. 78 (shaded areas). The shape of the background caused\nby these muons can be simulated. However, the exact number of µ+ which\nstop in the regions I and II depends strongly on the momentum distribution\nof the initial muon beam, which is not sufficiently well known.\n\nTherefore, the two measured distributions along the z-axis (for t = 150 ns\nand t = 2000 ns) are fitted simultaneously with the sum of 4 contributions:\n\n1. Background arising from the region I\n\n2. Background arising from the region II\n\n3. Linear (in time) background\n\n4. Simulation of all µ+ that stop in the gas (which includes µ+ in the\nactive region).\n\nThe shapes of these 4 contributions are known, under the assumption that\nall the detectors (S1 to S21 ) have the same detection efficiency. Each of the\n\n\n\n126 longitudinal compression\n\ncontributions has to be scaled with a different scaling factor to account for\nthe different stopping probabilities in region I, region II, the gas, and prompt\nstops at the target lateral walls. The additional linear background allows us\nto account for possible misalignment between the target and the magnetic\nfield axis, which would lead to different numbers of muon wall stops at the\npositions of the various scintillators.\n\nEven though we observe fair agreement between the measurement and\nthe simulation (reduced chi-square χ2red = 2.3, for 37 degrees of freedom),\nit is difficult to extract precise values of the compression efficiency and of\nthe width of the muon swarm from these measurements, given the limited\ngeometrical resolution of the detectors S1 to S21 and the large background\nfrom regions I and II. The relatively large χ2red could be attributed to small\nvariations of the detector efficiencies.\n\nIn order to better quantify the compression efficiency we turn our atten-\ntion to the two telescope detectors T1 & T2 in coincidence that were placed\nbelow the target, at z = 0, as shown in Fig. 68. These scintillators had di-\nmensions of 32× 3× 3 mm3 and were also read out by 3× 3 mm2 SiPMs.\nMassive brass shielding all around the target ensured that coincidence hits\nin T1 & T2 originated only from muons decaying within the small region\nbetween about z = ±3 mm in the center of the target, as shown in Fig. 77.\nFrom the time difference t = t1 − t0 between the positron hits in T1 & T2\n\nin coincidence (at time t1) and the entrance detector, at time t0, a time spec-\ntrum can be obtained as shown in Fig. 79. The time spectra were recorded\nfor different applied electric potentials.\n\nIt can be seen in Fig. 79 that if no electric field is applied (black points)\nthe number of muons decaying in the center of the target (in the T1 & T2\n\nacceptance region) stays constant (after compensation for µ+ decay). When\na negative potential (red points) is applied in the center of the target cell, the\nmeasured number of counts increases with time. In that case, the stopped\nmuons are attracted towards the potential minimum, so that more muons\ndecay within the acceptance region of T1 & T2. This means that the muon\nswarm extent has been decreased in the z-direction, representing a comple-\nmentary way to demonstrate longitudinal compression. On the contrary, if\nthe “wrong\" polarity is applied (green points), the muons drift away from\nthe center of the target, out of the acceptance region of T1 & T2. The very\nfew counts at late times in that case are due to some small background\n(mostly muons stopping in the wall of the target). The reduction to a nearly\nbackground-free measurement represents a major improvement compared\nwith the earlier measurements from 2011 [19] and compared to the measure-\nment of the µ+ z-distribution of Fig. 78.\n\nThe measurements of Fig. 79 are compared to Geant4 simulations. The\nthree simulations (+, − and 0 voltage) are fitted simultaneously to the corre-\nsponding measured time spectra. The simulated time-spectra are obtained\nby convolution the appropriate muon distribution as a function of time with\nthe position-dependent detection efficiency of T1&T2 in coincidence, accord-\ning to Eq. (208). For fitting, only two free parameters were used: a common\nscaling factor and a common flat background. The common scaling factor\n\n\n\n5.1 improved test of the longitudinal compression 127\n\nFigure 79: Measured (dots) and simulated (line) time spectra for 5 mbar He gas\npressure and potentials of −500 V (red), 0 V (black) and +500 V (green).\nThe data has been normalized to the number of incoming muons in the\nentrance detector D0 and fitted simultaneously with only 2 free parame-\nters, a common normalization and a common background. In total 5 · 108\nmuons have been simulated.\n\nis needed to remove the uncertainties related to the total positron detection\nand muon stopping efficiencies. The flat background was included in the\nfit to account for potential misalignment of the target with respect to the\nmagnetic field axis, which would lead to increased muon stops in the walls.\n\nThe simultaneous fit of the 3 curves has a reduced chi-square χ2red = 1.41\n(for 46 degrees of freedom). Introduction of additional losses during the\ncompression process, as detailed in the next section, improves the χ2red value\nto 0.95. Alternatively, a smaller χ2red can also be obtained by a minor tuning\nof the detector acceptance, related to uncertainties in the position, tilt and\nenergy threshold of the T1 & T2 scintillators, as will be explained later.\n\n\n\n128 longitudinal compression\n\n5.2 additional muon losses?\n\nAs mentioned in the introduction, in the previous (2011) experiment [19],\nthe data displayed lower-than-expected compression efficiency, which was\nvisible from the premature termination (at around t = 0.5 µs) of the com-\npression in the measured time spectra. This was attributed to severe losses\nof \"free” muons, hypothesized to have been caused by impurities present in\nthe helium gas (from outgassing of the Araldite glue and PCB boards used\nin the target construction). The presence of such impurities would cause\nthe capture of low-energy muons by the contaminant molecules. Addition-\nally, other issues affected the previous measurements such as misalignment\nbetween target and magnetic field axis, small target size and poor spatial\nresolution of the positron detector. These issues suppress the rise in the time\nspectra, thereby imitating the signature of the \"free\" muon losses, making it\ndifficult to quantify the actual compression efficiency.\n\nIn the experiments presented here, care was taken to improve the aspects\nlimiting the previous experiment: the detector resolution was enhanced by\nintroducing more shielding around the positron detectors and using the tele-\nscope detectors (T1 & T2) in coincidence instead of a single detector, the\ntarget size was increased and the electric field strength and homogeneity\nwere improved. As a consequence, the sensitivity to target misalignment\nwas considerably reduced. Additionally, high gas purity was ensured by\nconstructing the target from glass plates, using a low-outgassing glue and\npurifying the He gas in a cold-trap before feeding it into the target.\n\nIn order to prove the hypothesis that indeed the compression efficiency\nis lowered if contaminants are present in the gas, we introduced in a con-\ntrolled way various amounts of contaminant gases into the pure helium gas.\nThe results of this test are presented in the time spectra of Fig. 80. It is ob-\nserved that the compression stops at earlier times when contaminants are\nintroduced, and consequently the compression efficiency decreases (i.e., the\nnumber of muons that are brought to the center of target).\n\nThe sensitivity of the measurements of Fig. 79 to the muon loss mecha-\nnisms has been investigated assuming constant (in time) effective loss rates\nR during compression, that cause the “free\" muon population to decrease ac-\ncording to e−Rt. Various simulations have been performed with R ranging\nfrom R = 0 up to R = 0.5 µs−1. The time spectrum with an effective loss rate\nR = 0.5 µs−1 would correspond roughly to the measured time spectrum with\n0.01 mbar H2 contamination (blue dashed line in Fig. 80).\n\nThe simultaneous fit of the simulations for +, − and 0 voltages to the\ncorresponding measurements has been performed for each effective loss rate\nR (as in Fig. 79, but for R 6= 0). Figure 81 (top) shows the fitted time spectra\n(only for negative voltage) for several effective loss rates R. Notice that the\neffective muon losses reduce the number of counts in the -HV time-spectrum\nat late times (after ≈ 2 µs, when the compression is almost finished.)\n\nFor each of the effective loss rates R, the χ2red between the measurement\n(only −500 V data) and simulations has been calculated and plotted in\nFig. 81 (bottom). A parabola was then fitted to these points (green line).\n\n\n\n5.2 additional muon losses? 129\n\nFigure 80: Measured time spectra for 5 mbar pure He gas, and various admix-\ntures of contaminants: (black) no additional contaminants, (dashed blue)\n0.01 mbar H2, (dashed orange) 0.02 mbar H2, (dashed green) 0.01 mbar\nof O2. All data was normalized to the number of incoming muons.\n\nThe best agreement (minimum χ2red) between simulation and measurement\nis obtained for R = 0.14 µs−1, corresponding to an additional effective loss\nat t = 2 µs of 1 − e−0.14 µs−1·2 µs = 24%. The χ2red, min + 1 is obtained for\nR = 0.35 µs−1 corresponding to an additional loss at t = 2 µs of 51%. There-\nfore, we conclude that the total additional \"free\" muon loss after 2 µs is\n24+27−24%.\n\nHad these losses been caused by capture of the muons by gas impuri-\nties, the partial pressure of the impurities would have been ∼ 10−3 mbar 2.\nHowever, the purity of the He gas in the target was analyzed with a mass\nspectrometer and such impurity levels have not been observed. Thus, it is\nreasonable to conclude that the origin of these \"effective losses\" R cannot be\nattributed solely to the impurities.\n\nOther explanations, such as small deviations of the detector acceptance\n(spatial resolution) and of the cross sections from the values implemented\nin the simulations, are more favored. In the following subsection, we will\nconsider the possible impact of the small detector acceptance variation on\nthe measured time spectra.\n\ndetector acceptance variation We have seen that in the measured\ntime-spectrum, the number of counts at late times in the -HV measurement\nis slightly smaller than predicted by the Geant4 simulation. The smaller\n\n2 The relation between fitted R and impurity concentration is obtained by fitting the simulation\nfor various loss rates R to the measured time spectra of Fig. 80 with the additional 0.01 mbar\nH2 contamination.\n\n\n\n130 longitudinal compression\n\nFigure 81: (Top) Simulated time spectra for loss rates of R =\n\n(0, 0.125, 0.25, 0.333) µs−1 (lines). The −500 V measurement is rep-\nresented by square points. (Middle) Residuals for the time spectra from\nthe top figure. For simplicity, error bars are shown only for the curve\nwith the reduced χ2 = 0.95. For other curves error bars are of the similar\nsize. (Bottom) Reduced χ2 as a function of the loss rate R. Second order\npolynomial was fitted to these points (green line).\n\n\n\n5.2 additional muon losses? 131\n\nnumber of the detected counts at late times could be explained by minor de-\nviations between the simulated and the actual position-dependent detection\nefficiency (acceptance).\n\nThe detector efficiency could be changed by minor variations in the po-\nsitions of the two telescope detectors, T1 and T2. A change in the detector\nposition leads to a change in the total detection efficiency and a change of the\ndetector efficiency shape (z-position dependence, FWHM). The changes in\nthe total detection efficiency are not important in this case, as they influence\nall the time-spectra (+, 0, -) in the same way, and thus this kind of variation\nis already accounted for in the normalization fit parameter.\n\nHowever, the changes in the shape of the detector efficiency would most\nstrongly influence the -HV time-spectra. Indeed, the attractive potential\ncauses the muon swarm z-distribution to change rapidly with time. The\nresulting time-spectrum therefore strongly depends on the shape of the de-\ntector efficiency. On the other hand, the time-spectrum of the 0 V mea-\nsurement would not be affected at all, since the muon do not move after\nstopping in the gas. The +HV measurement is also only slightly affected by\nthe variation of the position-dependent detection efficiency, since only very\nfew muons actually stay in the detector acceptance region.\n\nLet us now study how the variation of the detection efficiency would in-\nfluence the number of counts in the T1 & T2 detectors in coincidence at\nlate times (where we observe the largest difference between simulations and\nmeasurements). According to Eq. (208), the number of counts in a small\ntime window (histogram bin) ∆t as a function of time with -HV applied is:\n\nN−(t) =\n∆t\n\nτ\n\n∫\nn−(z, t)ε(z)dz, (209)\n\nwhere n−(z, t) is the number of muons at position z at time t (for -HV) and\nε(z) is the position-dependent detection efficiency of the T1&T2 in coinci-\ndence. Similarly, for the 0 HV and the +HV, we can write:\n\nN0(t) =\n∆t\n\nτ\n\n∫\nn0(z, t)ε(z)dz (210)\n\nN+(t) =\n∆t\n\nτ\n\n∫\nn+(z, t)ε(z)dz, (211)\n\nwhere n0(z, t) and n+(z, t) are the numbers of muons at position z at time t,\nfor 0 HV and +HV, respectively.\n\nWe can note that, for 0 HV, the muon z-distribution (after stopping) is\nconstant in time and is the same as the muon stopping distribution in the\n-HV measurement, n−(z, t = 150 ns) (after correcting for muon decay). Thus,\nassuming the muon stopping distribution to be approximately uniform we\ncan write:\n\nn−(z, t = 150 ns) = n0(z, t) ≡ n0. (212)\n\n\n\n132 longitudinal compression\n\nThe detected number of counts at late times for the 0 HV measurement is\nthen simply3:\n\nN0,late =\n∆t\n\nτ\nn0\n\n∫\nεt(z)dz. (213)\n\nFurthermore, if we assume that when -HV is applied, all muons stopped in\nthe active region (between zmin = −100 mm and zmax = 100 mm) are at\nlate times at z = 0, we can write n−(z, t) at late times as:\n\nn−,late(z, t) = δ(z)\n\nzmax∫\nzmin\n\nn0 dz = δ(z)n0 · (zmax − zmin), (214)\n\nwhere δ(z) is the delta function and n0 · (zmax − zmin) is the total number\nof muons stopped in the active region. Thus, the number of counts N−(t) at\nlate times can be written as:\n\nN−,late =\n∆t\n\nτ\n\n∫\nn−,late(z, t) ε(z)dz (215)\n\n=\n∆t\n\nτ\nn0(zmax − zmin)\n\n∫\nδ(z)ε(z)dz (216)\n\n=\n∆t\n\nτ\nn0(zmax − zmin) ε(z = 0). (217)\n\nThe ratio r of counts between -HV and 0 HV measurements at late times,\nusing Eqs. (217) and (213) takes the form:\n\nr =\nN−,late\n\nN0,late\n=\n\n(zmax − zmin)ε(z = 0)\nzmax∫\nzmin\n\nε(z)dz\n\n. (218)\n\nGraphically, the ratio r is essentially the ratio of two areas defined in Fig. 82\n\n(left):\n\nr =\nA2\n\nA1\n. (219)\n\nThe behavior of the ratio r defined in this way was studied for small\nvariations of the telescope position, using Geant4 simulations of the positron\ndetection efficiency, similar to Fig. 77. An example of such study is shown in\nFig. 82 (right) where r is given as a function of distance d of the T1 scintillator\nfrom the target wall (glass plate), while keeping the T2 scintillator position\nfixed (the design value of d is 0). We can see that a change of d by only\n0.5 mm (which is within the mechanics tolerance) from the design position\nwould decrease the ratio r by about 5% (from 21.6 to about 20.8). This\nwould then translate to a 5% decrease of the number of counts at late times\nin the measured time-spectra of Fig. 81 (top), thereby imitating the signature\nof \"free\" muon losses. This 5% decrease in the number of counts at late\ntimes roughly corresponds to the decrease caused by an effective loss rate\nR = 0.125 µs−1 (compare red and black lines of Fig. 81 (top)). Thus, such a\nsmall variation of the detector efficiency could as well be responsible for the\nsmall “effective losses”, observed in the measurements.\n\n3 We have assumed that the background in the measurements is very small, i. e., that the\nnumber of counts in the +HV measurements at late times is almost zero (which is well\njustified, see Fig. 79).\n\n\n\n5.3 longitudinal compression with E×B drift 133\n\nFigure 82: (Left) Plot of T1&T2 detection efficiency as a function of z (green line).\nThe two areas A1 and A2, used to define the ratio r according to Eq. (219)\nare also indicated. (Right) Ratio r as a function of the distance d of the\nT1 detector from the target wall. The T2 position was kept fixed.\n\n5.3 longitudinal compression with E×B drift\n\nThe next important step towards the realization of the complete muCool\ndevice is the demonstration of the so-called E×B drift (also at room temper-\nature). This drift guides the muons from one compression stage to the next,\nand is also used to extract them finally into vacuum through a small orifice.\n\nFor this purpose we modified the longitudinal target cell to generate an\nelectric field with an additional vertical component Ey (see Fig. 83):\n\nE =\n\n\n0\n\nEy\n\n−Ez\nz\n|z|\n\n , (220)\n\nwhere Ey = 120 V/cm and Ez = 60 V/ cm. This Ey component leads to a\ndrift of the muons in the +x-direction, on top of the longitudinal compres-\nsion caused by the electric field in the z-direction. The electrodes produc-\ning the needed electric field are shown in Fig. 84, and the obtained electric\nfield and the electric potential simulated with COMSOL Multiphysics® are\nshown in Fig. 85. To construct the target with such electrodes, the electrodes\nfor all target walls were printed on a single Kapton foil, and then folded into\nthe rectangular shape, as sketched in Fig. 86.\n\nFor this measurement, target cell was enlarged to a transverse cross section\nof 24× 12 mm2 and, additionally, the muon beam was injected off-center, as\nindicated in Fig. 83 by the yellow circle. These two modifications allowed the\nµ+ to drift for longer times before hitting the right wall of the target, thus\nenhancing the sensitivity of the measured time spectra to the muon drift.\n\nIn total seven plastic scintillators (D1-D7) were mounted around the target\nat z = 0 along the x-direction to monitor this drift, as shown in the sketch of\nFig. 83. The scintillators had a cross section of 2× 2 mm2 and length (in the\n\n\n\n134 longitudinal compression\n\nFigure 83: Sketch of the setup to measure the E× B drift. The muon beam was in-\njected off-axis to allow the muons to drift a longer distance before hitting\nthe lateral wall. The electric field has a z-component for longitudinal com-\npression and a y-component to drift the muons in the x-direction. Seven\nscintillators (D1-D7) are positioned above the center of the target at z = 0,\nbut at different x-coordinates, to monitor the muon swarm movement in\nx-direction.\n\ny-direction) of 32.5 mm for the three central scintillators (D3, D4, D5) and\n48.5 mm for the four scintillators on the left and right sides of the target (D1,\nD2, D6, D7). As before, the scintillators were read out by 3× 3 mm2 SiPMs.\n\nThe simulated detection efficiency of these detectors is shown in Fig. 87.\nWe can notice that the detection efficiency depends on both x and z-coordinates.\nIn order to distinguish between Ê× B̂ drift and longitudinal compression, it\nwas thus advantageous to have relatively weak Ey field, so that the Ê× B̂\ndrift is so slow that it is also occurring after the longitudinal compression is\nfinished.\n\nThe simulated spatial distribution of the muon swarm as a function of\ntime is given in Fig. 88. This distribution, convoluted with the correspond-\ning position-dependent detection efficiencies of Fig. 87, gives rise to the\nsimulated time spectra shown in Fig. 89, together with the corresponding\nmeasurements.\n\nAt early times, the time spectra are dominated by the muon swarm com-\npression in the z-direction, thus the number of detected positrons increases\nin all scintillators. However, after about 2 µs, the number of detected positrons\nin the detectors on the left side of the target (D1-D3) decreases, indicating\nthat the muons are moving out of the acceptance region of this scintilla-\ntor. On the other hand, the detectors on the right side of the target (D5-D7)\ndetect increasingly more positrons. This finding indicates that the muon\n\n\n\n5.3 longitudinal compression with E×B drift 135\n\nFigure 84: Electrodes that define the electric field given by Eq. (220). The color\nscale shows the potential distribution on the electrodes (simulated with\nCOMSOL Multiphysics®).\n\nFigure 85: Simulated electric field direction (black arrows) and potential (colored\ncontours) in the xz-plane at y = 0 mm (top) and yz-plane at x = 0\n\n(bottom). In this simulation the potentials on the electrodes of Fig. 84\n\nwere used as boundary condition.\n\nswarm slowly drifts in the +x-direction towards the prospective point of\nextraction.\n\nAlso in this case, the measured time spectra of all seven “drift\" detectors\nwere fitted simultaneously with the simulation allowing for one common\nscaling factor, and a different flat background for each detector. A fair agree-\nment between data and simulation has been observed (χ2red = 3.44 for 433\n\n\n\n136 longitudinal compression\n\nFigure 86: Sketch of the electrodes required to achieve the electric field given by\nEq. (220). All the electrodes were first printed on a single Kapton foil\n(left) and then folded into the rectangular shape (right). The folding\ndirection is indicated by the arrows.\n\ndegrees of freedom), given that no systematic effects were accounted for in\nthe simulation.\n\nTo study the effect of additional muon losses on the measured time spec-\ntra, an effective loss rate R has been introduced in the simulation, analo-\ngously to the procedure described in the previous section. The obtained time\nspectra for the various R were then fitted to the data. The best agreement\nbetween simulation and the data is obtained for R = 0.125 µs−1, consistent\nwith the loss rate R reported in the previous section. The best fit, which\ngives a χ2red = 2.23, is shown in Fig. 89 (dashed lines).\n\nTo extract the uncertainty of the hypothetical loss rate that describes the\ndata the best, we repeated the procedure of the previous section: namely,\nfitting with a parabola the plot of reduced chi-square versus loss R rate\n(similar to Fig. 81). Note that for that procedure, the error bars of the data\nwere scaled up by a factor of 2.23 to produce a minimum reduced chi-square\nof 1. From this analysis we determined that the loss rate that best describes\nthe data is R = 0.17+0.23\n\n−0.17 µs−1 corresponding to a total \"free\" muon loss at\nt = 2 µs of 28+26−28%.\n\nEven with the additional muon losses introduced, some systematic dis-\ncrepancies between the data and the simulation still remain. Yet the main\ngoal, namely to demonstrate the feasibility of the E× B drift also at room\ntemperature, has been achieved.\n\nThe difference between the simulation and the measurement can be at-\ntributed either to the simplified modeling of the additional losses (without\nany energy dependence), a misalignment of the beam with respect to the\nmagnetic field axis or a small variation of the detector acceptances compared\nto the design values.\n\n\n\n5.3 longitudinal compression with E×B drift 137\n\nFigure 87: Projection of the position-dependent detector efficiency in the zx-plane\n(left column), along the x-axis (middle column) and along the z-axis\n(right column) for the seven detectors D1-D7 (see Fig. 83). These de-\ntectors were sensitive to the muon drift in x-direction, but also to the\nmuon compression in the z-direction. The colors in the plots in the left\ncolumn represent the probability to detect a positron from a muon decay\nas a function of the muon decay position.\n\n\n\n138 longitudinal compression\n\nFigure 88: Muon positions projected in the xz-plane for various times. The time is\ngiven by the color scale. Muon beam centered at x = −6 mm with 3 mm\nradius is stopped uniformly along the z-axis. The muons drift in the\n+x-direction while compression occurs in z-direction.\n\nAccording to the simulation, the drift velocity is about 2 mm/µs. This\nvalue can be increased in the final setup by increasing the strength of the\nelectric field in the y-direction.\n\n5.4 conclusions\n\nThe longitudinal compression stage of the muCool device under develop-\nment at PSI has been demonstrated. An elongated muon swarm of 200 mm\nlength has been compressed to below 2 mm length within 2 µs. Good agree-\nment between the simulation and the measurement has been observed.\n\nFurthermore, the ability to drift the µ+ beam in E× B-direction towards\nthe prospective position of the extraction hole has been demonstrated by\nperforming a measurement with the electric field having also a component\nperpendicular to the magnetic field.\n\nIn both cases, slightly better agreement between simulations and measure-\nments is achieved by including small additional effective losses in the simula-\n\n\n\n5.4 conclusions 139\n\nFigure 89: Measured (dots) time-spectra for the seven scintillators D1-D7. The in-\ncrease and decrease of positron counts in the detectors on the right and\nleft side, respectively, indicates that the muons are moving towards the\nright. Simulated time-spectra without any additional muon losses (con-\ntinuous lines) and with an additional loss rate of R = 0.125 µs−1 (dashed\nlines) are fitted to the measured time spectra. The χ2red for R = 0 is 3.44\nfor 433 degrees of freedom. Introducing the additional loss rate R = 0.125\nµs−1 improves the reduced χ2 to 2.23. Note that the data and the simula-\ntions have been corrected for the finite muon lifetime by multiplying the\ncounts with et/2197 ns.\n\ntion, that decrease the actual compression efficiency by about 25% relative to\n(ideal) simulations. These effective losses could be caused by gas impurities.\nAlternatively, the small discrepancies between the simulations and measure-\nments could be attributed to a minor (5%) overestimation of the detector\nacceptance or minor variation of the cross sections for muonium formation,\n\n\n\n140 longitudinal compression\n\nmuonium ionization and muonium-He elastic scattering. In any case, even\nwith this additional effective loss, the proposed µ+ compression efficiency\nof 10−3 for the full muCool device is attainable.\n\n\n\n6\nM I X E D T R A N S V E R S E - L O N G I T U D I N A L C O M P R E S S I O N\n\nAfter having successfully demonstrated the muon transverse and longitudi-\nnal compression independently, as presented in Chapters 4 and 5, the next\nstep is to combine these two stages together. In the original scheme [16] the\nlongitudinal compression stage (at room temperature) follows the transverse\nstage (at cryogenic temperature), as shown in Fig 2. However, in this thesis\nwe have developed an alternative approach of combining these two stages\ninto a single stage at cryogenic temperatures, where the so-called mixed\ncompression occurs. i. e., compression in both longitudinal and transverse\ndirections at the same time. This approach would simplify the beam-line\ndevelopment significantly and provide us with a simple setup to test one of\nthe major challenges of the muCool device: the extraction into vacuum.\n\nThe working principle and the most relevant design considerations of\nsuch a stage with mixed transverse and longitudinal compression will be\nexplained in this chapter. The results of the preliminary tests of the mixed\ncompression stage obtained in the 2017 beam-time will also be presented\nhere.\n\n6.1 working principle\n\nFigure 90: Sketch of the mixed transverse-longitudinal compression setup. The ver-\ntical (y-direction) density gradient and the Ex and Ey electric field com-\nponents are the same as in the transverse compression setup. However,\nin this case, the electric field also has a component in the z-direction. This\ncomponent points towards the target mid-plane at z = 0, similarly to the\nlongitudinal compression stage.\n\nThe mixed transverse-longitudinal compression can be achieved by slightly\nmodifying the electric field shapes of the transverse stage, while keeping the\n\n141\n\n\n\n142 mixed transverse-longitudinal compression\n\nsame temperature (density) gradient. The x− and y− components of the\nelectric field remain the same as in the transverse stage (Ex = Ey), but an\nadditional z-component of the field is introduced, pointing towards the tar-\nget center at z = 0, similar to the longitudinal stage. A sketch of the electric\nfields and the target geometry leading to a simultaneous compression of the\nmuon beam in both transverse and longitudinal direction is shown in Fig. 90.\n\nTo illustrate such a mixed compression, consider the muon trajectories\nof Fig. 91 simulated in Geant4. In this simulation the following condi-\ntions were assumed: a temperature gradient of 6.0 K-18.6 K at 8.5 mbar,\nB = 5 T and electric field E = (0.9, 0.9, 0.45) kV/cm for z < 0 and E =\n\n(0.9, 0.9,−0.45) kV/cm for z > 0. The simulation was done starting from\nµ+ of 1 eV energy at x = −20 mm, homogeneously distributed within\ny ∈ [−9, 9] mm and z ∈ [−20, 20] mm. The target dimensions were the\nsame as for the transverse compression target of Chapter 4.\n\nThe left plot of Fig. 91 shows the projection of the muon trajectories in\nthe yx-plane. This plot exposes the transverse compression of the muon\nbeam during the drift in the x-direction (y vs. x evolution): muons start at\nx = −20 mm with 18 mm spread in y-direction; within 2.5-5 µs, they reach\nx = 20 mm while having their vertical spread reduced to 0.33 mm (FWHM).\nThe projection of the muon trajectories in the zx-plane is shown in Fig. 91\n\n(right). This plot depicts the evolution of the muon z-distribution vs. x, i. e.,\nit exposes the longitudinal compression: muons starting with a spread of\n40 mm in the z-direction have a spread of only 0.28 mm at x = 20 mm.\n\nFigure 91: Geant4 simulation of the muon trajectories under the following con-\nditions: temperature gradient of 6.0 K-18.6 K at 8.5 mbar, B = 5 T\nand electric field E = (0.9, 0.9, 0.45) kV/cm for z < 0 and E =\n\n(0.9, 0.9,−0.45) kV/cm for z > 0. The projections of the muon trajec-\ntories in the yx-plane (left) and in the zx-plane (right) are shown, high-\nlighting the transverse and longitudinal compression, respectively. The\n1 eV muons start at x = −20mm with 18 mm spread in the y-, and 40mm\nspread in the z-direction. Within 4.5 µs they reach the tip of the target at\nx = 20 mm, where their spread is 0.33 mm in the y-, and 0.28 mm in the\nz-direction.\n\nThe muon z- and y-distributions at x = 20 mm are shown in Fig. 92 (left\nand middle, respectively). The right panel of Fig. 92 shows the distribution\nof the time muons needed to reach x = 20 mm. This arrival (compression)\ntime distribution is relatively wide because muons starting at x = −20 mm,\n\n\n\n6.2 electric field design 143\n\nbut at different y-positions travel through different gas densities and thus\nhave different drift velocities.\n\nFrom these distributions the compression efficiency (at x = 20 mm) can be\npredicted: neglecting µ+ decay, 80% of the muons that start at x = −20 mm\nare compressed into an area of ∆y× ∆z = 1× 1 mm2 at the x = 20 mm\nplane. The remaining 20% of the muons are lost through muonium forma-\ntion. When the µ+ lifetime is included, the compression efficiency is reduced\nfrom 80% to 20%.\n\nFigure 92: Muon z-distribution (left), y-distribution (middle) and arrival time distri-\nbution (right) at x = 20 mm. These distributions were extracted from the\nsimulation of Fig. 91.\n\nThis scheme could thus lead to a larger total compression efficiency than\nthe one in the original proposal [16], because the compression time is re-\nduced from about 4.5− 7 µs to about 2.5− 5 µs. The compression time is\nreduced because two intermediate stages1 of the original proposal have been\nremoved, thus shortening the length of the device in the x-direction.\n\n6.2 electric field design\n\nAs introduced above, the electric field needed to obtain the mixed transverse-\nlongitudinal compression takes the form:\n\nE =\n\n\nEx\n\nEy\n\n−Ez\nz\n|z|\n\n , (221)\n\nwhere Ex, Ey and Ez are constants (with Ex = Ey typically, as in the trans-\nverse compression stage). Such an electric field has to be defined inside\nthe target volume of triangular transverse shape. This is done by lining the\ntarget walls (volume boundaries) with electrodes of appropriate shape and\nsetting them at the correct electric potentials (different HV). The required\nelectrode shape and HV can be determined from the electric potential that\nproduces the desired electric field. Such an electric potential is obtained by\n\n1 The transition regions between the transverse and longitudinal stages, and between the lon-\ngitudinal and extraction stages.\n\n\n\n144 mixed transverse-longitudinal compression\n\nintegrating the desired electric field E, given by Eq. (221), with respect to an\narbitrary reference point O:\n\nV(r) = V(O) −\n∫ r\n\nO\n\nE · dl. (222)\n\nWe choose O to be at the origin of the coordinate system (0, 0, 0), placed at\nthe target center (see Fig. 91). The potential at position r = (x,y, z) can then\nbe written as:\n\nV(x,y, z) = V(O) −\n∫ r\n\nO\n\nE(r′) · (dx′x̂ + dy′ŷ + dz′ẑ) (223)\n\n= V(O) −\n\n∫ r\n\nO\n\n(Exx̂ + Eyŷ − Ez\nz′\n\n|z′|\nẑ) · (dx′x̂ + dy′ŷ + dz′ẑ)\n\n(224)\n\n= V(O) − Exx− Eyy+ Ez |z| , (225)\n\nby inserting the electric field of Eq. (221) in the last two lines.\nThe potential of Eq. (225) is shown in Fig. 93 (top) for Ex = Ey = 3Ez,\n\nassuming that the ground (V = 0) is at the tip of the target and the maximum\napplied HV is 1 kV. Typically, it suffices to define the potential at the top, side\nand bottom walls of the target, because the end-caps are relatively small\nand far away from the region of interest (“active region”). The potential at\nthese walls is depicted in Fig. 93 (middle). By selecting the equipotential\nlines spaced approximately every 25 V, we obtain the electrodes of Fig. 93\n\n(bottom).\nTo construct this target of triangular shape, we line a Kapton foil with\n\nall the required electrodes (see Fig. 94), and then we fold the foil in the\ntriangular shape as in Fig. 93 (bottom). Each two neighboring electrodes are\nconnected via 100 MΩ resistors, acting as voltage dividers. By applying HV\nat the bottom corner of the target and by grounding the last electrodes at the\ntip of the target, the desired potential distribution is achieved.\n\nNote that the angles and the spacing of the electrodes depends on the tar-\nget walls considered, as visible in Fig. 94. This is because different target\nwalls intersect the equipotential planes differently. Since we know the exact\nanalytical form of the electric potential (Eq. (225)), and the shape of the tar-\nget, the spacing and the electrode angles can also be determined analytically,\n(see Appendix D). The result of the calculation carried out there gives us the\nfollowing electrode angles:\n\ntanα = (cos θ− sin θ)\nEx\n\nEz\n, (226)\n\ntanβ =\nEx\n\nEz\n, (227)\n\ntanγ = (cos θ+ sin θ)\nEx\n\nEz\n, (228)\n\nwhere α, β and γ are the electrode angles at top, side and bottom target\nwalls, respectively, as defined in Fig. 94. The θ is the target opening angle\n(see Fig. 93 (top)). We have assumed here that the target is symmetric with\n\n\n\n6.2 electric field design 145\n\nFigure 93: Scheme of the procedure for designing the electrodes that produce the\nelectric field needed for the mixed compression (E = (Ex,Ey,−Ez z|z| )).\nFirst, the electric potential in the target volume corresponding to such\nan electric field is calculated analytically, using Eq. (225). The top panel\nshows several equipotential surfaces (spaced by 50 V), assuming maxi-\nmum applied HV of 1 kV at the bottom corner and V = 0 at the tip of\nthe target. The potential at the target walls is shown in the middle panel.\nBy selecting equipotential lines with about 25 V drop between them, we\nobtain the electrodes shown in the bottom panel. Such electrodes ap-\nproximately give rise to the desired electric field when set to the correct\npotentials.\n\n\n\n146 mixed transverse-longitudinal compression\n\nrespect to the y = 0 plane and that Ex = Ey. These angles depend on the\nratio Ex/Ez of the x- and z- components of the electric field, but also on the\ntarget geometry (θ). Typically γ > β > α. Note that the electrode spacing is\nproportional to the cosine of these angles.\n\nFigure 94: The electrodes of Fig. 93, prior to the folding of the Kapton foil. The\nelectrodes have different angles (α, β, γ) and spacing on the different\ntarget walls (top, side, bottom).\n\nThe electrode shapes obtained in such a way are not well suited for the\npractical realization. The electrode lining has to be optimized to smooth\npossible local electric field enhancements (which could lead to electric break-\ndown), to account for the mechanical constraints and also to maximize the\ncompression efficiency. For this purpose several modifications have to be\nmade, starting from the lining of Fig. 94; the final result is shown in Fig. 95.\nThe aspects that were optimized2 are described and motivated in the follow-\ning:\n\nelectrode shapes : All the sharp electrode corners have been rounded\n(particularly at z = 0 and at the edges of the active region) to avoid\nlocally enhanced electric fields.\n\nelectrode width and spacing : To minimize the electric field between\ntwo strips3, the ratiow/d of electrode width to electrode spacing needs\nto be minimized. This can be done by decreasing w and/or increasing\nd. However, there are some limitations: w < 0.1mm is difficult to man-\nufacture; large gaps (d−w � 1 mm) would lead to distortions in the\nfield shape inside the target; for simplicity, d should match standard\n\n2 Note that for these optimizations the target dimensions were fixed to be the same as the\nprevious transverse target.\n\n3 The potential difference ∆V between two strips is proportional to the center-to-center dis-\ntance d between two strips. However, the electric field between two strips is inversely pro-\nportional to the gap (d−w) between two strips: E = ∆V/(d−w) ∝ 1/(1−w/d) , where w is\nthe electrode width. For large w/d the electric field between two strips is larger than in the\ntarget volume.\n\n\n\n6.2 electric field design 147\n\nSMD resistor sizes. As a result of these constraints, typically we used\nw = 0.4 and d = 1.4 mm.\n\nelectrode placement : The electrodes that would cross the top corner\nof the target have been removed. The reason behind this is that the\nelectrode material (Ni+Au) is relatively brittle so that a sharp folding\nof the Kapton foil could break those electrodes. Furthermore, the elec-\ntrodes that would cross the bottom corner were merged into a single\nmetallic surface. This is beneficial for the electric breakdown, because\nthe maximum HV that we have to apply inside the target is reduced.\nEven with this simplification, the electric field in the region of interest\nis sufficiently well defined, as confirmed by Comsol simulations (see\nFig. 97).\n\nEx/Ez ratio : The strength of the Ez component (relative to the Ex compo-\nnent) is also constrained. On one hand, if Ez is too weak, the longitu-\ndinal compression will not be very efficient. On the other hand, if Ez\nis too strong, this can lead to electric breakdown. There are few other\n(unwanted) consequences of having very strong Ez:\n\n• The Maxwell’s equation ∇ · E = 0 constrains the x- and y- electric\nfield components around z = 0, where the z-component of the\nfield suddenly changes the direction. Therefore, around z = 0\n\nthe electric field in the xy-plane (needed for transverse compres-\nsion) will deviate from the required 45° field. This deviation only\nbecomes important for relatively strong Ez (Ez � Ex). Further-\nmore, the compression efficiency decreases for stronger Ez, be-\ncause muons reach the target mid-plane (at z = 0) before the\ntransverse compression is completed. The non-ideal shape of the\nelectric field at z = 0 reduces thus the efficiency of the transverse\ncompression.\n\n• The transverse compression of the muons depends strongly on\nthe average muon energy. If the muons are accelerated strongly\nin the z-direction, the total energy of the muons increases so that\nthe muons will simply drift in the E×B direction at all densities,\nand not just at the low gas density. This effect can be counteracted\npartially by slightly decreasing the strength of the Ex and Ey field\ncomponents compared to the “pure\" transverse compression.\n\nTypically, the Ex/Ez ratio is constrained to be between 1 and 4 for\nthe above listed reasons. Ideally, the ratio Ex/Ez should be such that\ntransverse and longitudinal compression finish around the same time.\n\nactive region length : The length of the active region in the z-direction\nis limited by the onset of the electrical breakdown. For example, for\nEz = 0.3 kV/cm (typical value), the reasonable active region length\nwould be ≈ 50 mm. At these conditions, muons starting at the periph-\nery of the active region reach the mid-plane (z = 0) within 5 µs, i. e.,\nwithin a time the transverse compression is completed. Beside break-\ndown issues, an increase of the active region length would also increase\n\n\n\n148 mixed transverse-longitudinal compression\n\nthe time muons need to reach the mid-plane (from the periphery), thus\nreducing the compression efficiency.\n\nFigure 95: Scheme of the electrode lining of the unfolded Kapton foil, optimized to\nprevent electrical breakdown and to account for mechanical constraints.\nThis design was developed by modifying various aspects of the elec-\ntrodes of Fig. 94, as explained in the text. To obtain the desired elec-\ntric field, we apply (fix) appropriate voltages to five selected electrodes\n(similar to the transverse compression target, see for example Fig. 42 in\nChapter 4). On the remaining electrodes the potential (color scale) is\ndistributed using three voltage dividers and computed using COMSOL\nMultiphysics®.\n\nThese optimizations were studied using the COMSOL Multiphysics® pack-\nage in the following way: first the potential distribution on all the electrodes\nis simulated in 2D (for the lining of Fig. 95). In this simulation, potentials\nof only 5 electrodes are fixed to the value indicated in Fig. 95. These values\nwere determined using Eq. (225). The potentials on the other electrodes are\nthen calculated using the COMSOL Multiphysics® “Electric currents” mod-\nule. Finally, the results of the 2D simulation are mapped to the walls of the\ntarget in 3D as a boundary condition (Fig. 96) and the potential in the tar-\nget volume is calculated using the COMSOL Multiphysics® “Electrostatics”\nmodule.\n\nThe resulting electric field lines (black lines) and electric potential distri-\nbution (color) computed with Comsol are given in Fig. 97, for two different\nplanes: yx-plane at z = 10 mm and zx-plane at y = 0. For reference, the di-\nrection of the initially assumed (“design”) electric field E = (Ex,Ey,−Ez z|z|),\nthat we used as a starting point to determine the electrode shapes, is also\nshown (gray lines).\n\n\n\n6.2 electric field design 149\n\nComparing the simulated with the “design\" electric fields, we can notice\nsome small differences. The main reasons for these small differences are\nrelated to the discretization of the potential at the target walls, resulting\nfrom the lining process, and from defining the potential only at the top, side\nand bottom target walls (for simplicity, we have not defined the potential at\nthe end-caps and inside the target volume). Furthermore, the requirement\nthat ∇ · E = 0 leads to a small distortion of the electric field shape at z = 0\n\n(where the z-component of the electric field changes direction). Despite these\ndeviations, the electric field is sufficiently similar to the “design” electric\nfield to achieve efficient mixed transverse-longitudinal compression, as will\nbe demonstrated with Geant4 simulations in the next subsection.\n\nFigure 96: Boundary conditions for the electric field simulation inside the target.\nThese boundary conditions are obtained by mapping the potentials from\nthe 2D simulations of Fig. 95 to the walls of the target. The colored strips\nrepresent the various electrodes that are set at various potentials.\n\nA target with the electrodes and dimensions of Fig. 96 was realized ex-\nperimentally. Figure 98 shows the photo of the lined Kapton foil before\nfolding and Fig. 99 shows the completed target. This target was then cooled\ndown to cryogenic temperatures and the HV was applied. Experimentally\nwe have reached a maximum HV of 4.48 kV (applied at the target bottom\ncorner), with temperature gradient from 5.1 K to 17.7 K at 8.7 mbar. Prior to\nthe discharge at 4.48 kV, the target was operated for one day at 4.3 kV, corre-\nsponding to an average electric field with Ex = 0.8 kV/cm, Ey = 0.75 kV/cm\nand Ez = 0.22 kV/cm.\n\ngeant4 simulation Optimization of the electrode design, as outlined\nin the previous section, needs to be done in conjunction with the Geant4\nsimulation of the muon trajectories to ensure that the compression efficiency\nis not compromised by the changes of the electric field design.\n\n\n\n150 mixed transverse-longitudinal compression\n\nFigure 97: Electric field lines (black lines) and electric potential (color) in the yx-\nplane at z = 10 mm (top) and zx-plane at y = 0 (bottom), simulated\nassuming the potential on the electrodes of Fig. 96 as a boundary condi-\ntion. For comparison, the “design” field lines with Ex = Ey = 3Ez, which\nwas the starting point for the electrode design, is represented by the gray\nlines.\n\nThe simulated muon trajectories using the optimized electric field of Fig. 97\n\nare shown in Fig. 100. The maximum applied HV was assumed to be 4.8 kV,\nproducing an electric field with Ex = 0.89 kV/cm, Ey = 0.84 kV/cm and\nEz = 0.24 kV/cm on average within the target volume. The density gradi-\nent was also simulated in COMSOL and imported in the Geant4 simulation,\nassuming temperatures from 6.0 to 18.6 K at 8.5 mbar.\n\n\n\n6.2 electric field design 151\n\nFigure 98: Photo of the electrodes of Fig. 95 printed on a Kapton foil. The voltages\napplied at some selected electrodes corresponding to the corners of the\ntarget are also shown. These values represent the maximum HV that we\nwere able to apply without onset of electrical breakdown.\n\nFigure 99: Photo of the folded target, starting from the Kapton foil of Fig. 98. Also\nvisible in the photo are the top and bottom sapphire plates, used to define\nthe vertical temperature gradient inside the target and one of the gas\ntubes, used to fill and circulate the He gas.\n\n\n\n152 mixed transverse-longitudinal compression\n\nSimilar as before, stopped muons of 1 eV energy start at x = −20 mm\nwith 18 mm spread in the y-direction and 40 mm spread in the z-direction.\nWithin 2-5 µs, the muons reach the plane at x = 17 mm. At this point, their\nextension is reduced to about 0.7 mm in the y-direction and to about 6 mm\nin the z-direction (see distributions of Fig. 101).\n\nThe probability that the muons reach the plane at x = 17 mm is 86.6%\n(without muon decay) and 21% (with muon decay). However, due to the\nslightly too weak Ez component (for a target length of 40 mm in the x-\ndirection), the longitudinal compression is sub-optimal, so that within an\n1× 1mm2 area at x = 17mm we have only 20% of the initial muons (without\nmuon decay) and 5% (with muon decay). The total compression efficiency\nis reduced compared to \"ideal\" case presented in the introduction. In order\nto try to increase this compression efficiency, for the beam time 2017, sev-\neral adaptations to the target were adopted, as explained in the following\nsection.\n\nFigure 100: Projections of the muon trajectories in the yx-plane (left) and in the\nzx-plane (right), simulated using Geant4 and assuming the simulated\nelectric field of Fig. 97 (with maximum HV = 4.8 kV) and the simulated\ndensity gradient of Fig. 41 (6-18.6 K at 8.5 mbar) and 5 tesla magnetic\nfield. Muons with 1 eV energy start at x = −20 mm with 18 mm spread\nin the y-direction, and 40 mm spread in the z-direction. Within 2-5 µs\nthey reach the tip of the target at x = 17 mm, where their spread is\n0.7 mm in the y- and 6 mm in the z-direction.\n\n6.3 measurements and results of the 2017 beam-time\n\nWith the target design described in the previous section, the longitudinal\ncompression was not efficient enough. There are two possible approaches to\nimprove the longitudinal compression: either increasing the z-component of\nthe electric field or increasing the length of the target in x-direction. In 2017,\nincreasing the Ez significantly was not feasible because of issues with the\nelectric breakdown. Therefore, we opted for changing the target geometry\nby increasing the length in x-direction from about 40 mm to 60 mm. At the\nsame time, we reduced the height of the target from 23.5 mm to 14 mm. This\nchange allowed us to keep the maximum applied HV relatively low, despite\nincreasing the length in x-direction by 50%. Moreover, this target geometry\nimproves the spatial resolution of the detectors used to monitor the muon\n\n\n\n6.3 measurements and results of the 2017 beam-time 153\n\nFigure 101: Muon z-distribution (left), y-distribution (middle) and arrival time dis-\ntribution (right) at x = 17 mm plane. These distributions were extracted\nfrom the simulation of Fig. 100. The shaded regions show the fraction\nof muons within a ±0.5 mm interval centered at the mean of z- and\ny-distributions. Due to large spread in the z-direction only 23% (ne-\nglecting muon decay) of the muons that reach the plane at x = 17 mm\nare within an 1× 1 mm2 area.\n\nmotion (because of the smaller average distances between the detectors and\nthe muons).\n\nSuch a target is shown in Fig. 102, along with the values of the electric\npotential at the target walls (from Comsol simulation). A photo of the actual\ntarget is given in Fig. 112.\n\nGeant4 simulations of the muon trajectories have been repeated for the\nnew target geometry in order to find the optimal conditions (electric field\nstrength, density gradient) for the compression. As before, the electric field\nand the density gradient simulated in Comsol were imported in the Geant4\nsimulation. The temperature gradient was assumed to be from 6.0 K to\n19.0 K at 8.75 mbar pressure. The Geant4 simulations were done for several\nvalues of the maximum HV.\n\nIn the simulations we start with muons of 1 eV energy, uniformly dis-\ntributed inside a cylinder oriented along the z-axis and centered at (−25, 0, 0) mm\nwith radius of 2.5 mm and length of 55 mm in z-direction. This corresponds\nto the stopping volume (in the active region) of the muons when they are\ninjected through an aperture of 2.5 mm radius.\n\nFigure 103 shows projections of the muon trajectories in the yx- and zx-\nplanes for HV = 6.0 kV. At this HV, muons need on average 4.6 µs to reach\nthe plane at x = 22 mm. During this time the muon spread in y-direction is\nreduced from 5 mm to 0.33 mm (FWHM) and the spread in the z-direction\nis reduced from 55 mm to 0.76 mm. The muon y, z and arrival time distribu-\ntions at x = 22 mm are shown in Fig. 104. From these distributions, various\nmuon compression efficiencies can be extracted for different HV values, as\nsummarized in Table 4. Following efficiencies are reported in the table:\n\n• ε: the percentage of muons that reach the plane at x = 22 mm, ne-\nglecting losses related to muon decay (i. e., assuming infinite muon\nlifetime),\n\n\n\n154 mixed transverse-longitudinal compression\n\nFigure 102: Sketch of the target used in the 2017 beam-time. The electrodes were\ndesigned to produce the electric field with Ex/Ez = 2.5 in the active\nregion extending from −27.5 to 27.5 mm in the z-direction. The colors\nrepresent the potentials at the target walls (from a COMSOL simulation).\n\n• ετ: the percentage of muons that reach the plane at x = 22 mm, ac-\ncounting also for losses caused by muon decay,\n\n• ε1×1: the percentage of muons that reach the plane at x = 22 mm and\nare within a 1× 1 mm2 area, neglecting losses related to muon decay,\n\n• ε1×1,τ: the percentage of muons that reach the plane at x = 22 mm\nand are within a 1× 1 mm2 area, accounting also for losses caused by\nmuon decay.\n\nForm the Table 4, we can see that ε decreases with increasing HV. This\nis caused by the losses of muons through muonium formation: indeed, the\nmuonium formation increases with the average kinetic energy, which in-\ncreases with HV. However, all other efficiencies (ετ, ε1×1, ε1×1,τ) increase\nwith increasing HV due to shorter compression times and reduced muon\nspread, especially in the z-direction.\n\nTo summarize, it is advantageous to have larger HV, because shorter com-\npression times lead to reduction of the most dominant muon loss mecha-\nnism: muon decay. Muon losses through muonium formation have much\nsmaller impact on the total efficiency.\n\nHowever, in the beam-time 2017 we had problems with the electrical break-\ndown. The maximum HV at which we could operate the target was 3.25 kV.\nTo compensate for the weaker electric field, the temperature gradient was\n\n\n\n6.3 measurements and results of the 2017 beam-time 155\n\nFigure 103: Projections of the muon trajectories in the yx-plane (left) and in the\nzx-plane (right). The trajectories were simulated in Geant4 assuming\nthe simulated electric field produced by the electrodes of Fig. 102 (with\nmaximum HV = 6.0 kV), the simulated density gradient (6-19 K at\n8.75 mbar) and a 5 tesla magnetic field.\n\nFigure 104: Muon z-distribution (left), y-distribution (middle) and arrival time dis-\ntribution (right) at x = 22 mm plane. These distributions were extracted\nfrom the simulation of Fig. 103. The shaded regions show the fraction\nof muons within a ±0.5 mm interval centered at the mean of the z- and\ny-distributions. About 64% (neglecting muon decay) of the muons are\nwithin an 1× 1 mm2 area.\n\nchanged to 9.2-30.8 K from the usual 6.0-19.0 K. The pressure was kept the\nsame (8.75 mbar) as in the simulation of Figs. 103 and 104.\n\nThe results of the Geant4 simulations under such conditions are shown in\nFig. 105 and 106. We can see that, even at these conditions, both transverse\nand longitudinal compression still occur, though slower, and thus with lower\ncompression efficiencies (see Table 4). Indeed, the compression efficiencies\nthat account also for losses due to muon decay (ετ and ε1×1,τ) are decreased,\nwhile the efficiencies without muon decay (ε and ε1×1) are increased com-\npared to the “ideal” case of Fig. 103. Such behavior is related to the too weak\nelectric field, which implies slower drift velocity. On one hand, the slower\ndrift velocity leads to longer compression times, thus decreasing the efficien-\ncies ετ and ε1×1,τ; on the other hand, the muonium formation decreases\nwith decreasing kinetic energy (see Fig. 22). Hence, the efficiencies ε and\nε1×1 are slightly larger than in the “ideal” case.\n\nTo test the mixed compression at these conditions, we injected 2 · 104 µ+/s\nat 12.5 MeV/c momentum. The setup used is largely similar to the setup\nused to measure the transverse compression and will not be described in\n\n\n\n156 mixed transverse-longitudinal compression\n\nHV Tlow Thigh(K) ∆y ∆z tcomp\nε ετ ε1×1 ε1×1,τ\n\n(kV) (K) (K) (mm) (mm) (µs)\n\n5.25 6.0 19.0 0.42 1.00 4.77 94.8% 10.9% 55.8% 6.4%\n\n5.50 6.0 19.0 0.34 1.00 4.71 94.5% 11.2% 56.8% 6.6%\n\n5.75 6.0 19.0 0.36 0.80 4.66 93.4% 11.4% 57.1% 6.7%\n\n6.00 6.0 19.0 0.33 0.76 4.62 92.6% 11.5% 59.6% 7.0%\n\n3.25 9.2 30.8 0.44 0.68 6.66 99.5% 4.9% 60.6% 2.9%\n\nTable 4: Summary of the beam sizes and compression times relevant for quantifying\nthe compression efficiency (at the x = 22 mm plane), for various target\nconditions (HV, temperature gradient). These numbers were extracted from\nthe Geant4 simulations (with 105 simulated muons). The quantities ∆y and\n∆z are FWHM of muon y- and z- distributions at x = 22 mm. The time\ntcomp is the average compression time (the time it takes muons to reach\nx = 22 mm, starting from about x = −25 mm). The efficiencies ε, ετ, ε1×1\nand ε1×1,τ are defined in the text.\n\nFigure 105: Projections of the muon trajectories in the yx-plane (left) and in the\nzx-plane (right). The trajectories were simulated in Geant4 using the\nsimulated electric field produced by the electrodes of Fig. 102 (with max-\nimum HV = 3.25 kV) and the simulated density gradient (9.2-30.8 K at\n8.75 mbar) and 5 tesla magnetic field. These are the conditions achieved\nduring the 2017 beam-time.\n\ndetail. Yet, there are two main differences (compared to the transverse com-\npression setup) that are worth highlighting. First, in this test we refrained\nfrom distinguishing between “pure drift\" and compression in the transverse\ndirection, as it was already demonstrated with a separate measurement. For\nthe same reason, only the measurements with temperature gradient were\nperformed. This choice allowed us to use only a single aperture of 2.5 mm\nradius to define the muon beam entrance position inside the target. Using\ntwo apertures (“top\" and “bottom\" holes), as before, would be difficult be-\ncause of the smaller target height (in the y-direction).\n\nMore importantly, the detection system was modified to expose both lon-\ngitudinal and transverse compression at the same time. A sketch of the\nvarious detector positions is shown in Fig. 107. There are two groups of de-\ntectors: the detectors used to monitor the transverse compression (Trans1-3)\nand the detectors used to monitor the longitudinal compression (Tiles1-6).\n\n\n\n6.3 measurements and results of the 2017 beam-time 157\n\nFigure 106: Muon z-distribution (left), y-distribution (middle) and arrival time dis-\ntribution (right) at x = 22 mm plane for the 2017 beam-time conditions.\nThese distributions were extracted from the simulation of Fig. 105. The\nshaded regions show the fraction of muons within a ±0.5 mm interval\ncentered at the mean of the z- and y-distributions.\n\nThe transverse compression detectors are quite similar to the ones used in\nthe beam-time 2015. The acceptance of these detectors is shown in Fig. 108.\n\nThe detectors for monitoring the longitudinal compression were placed in\ntwo rows and read-out in coincidence to improve the spatial resolution. We\nhad three pairs of longitudinal detectors in coincidence: upstream (Tiles1 &\nTiles4), middle (Tiles2 & Tiles5) and downstream (Tiles3 & Tiles6). The up-\nstream and downstream tiles were placed above the periphery of the active\nregion (z = ±22 mm), while the middle tiles were placed above the center\n(z = 0). All of them were placed above the tip of the target (at x = 20 mm)\nin order to be sensitive to muon z-distribution only at late times, when the\nlongitudinal compression is (almost) finished.\n\nThe acceptance in the zx-plane of these three tile pairs in coincidence is\nshown in Fig. 109. From these acceptance plots, we can notice that the tiles\nare also sensitive to a muon drift in the x-direction. For this reason, three\npairs of detectors are necessary: if we had only the middle one, we would\nnot be able to distinguish between drift in the x-direction and compression\nin the z-direction. In other words, a comparison of the time-spectra of the\nmiddle tiles with upstream and downstream tiles can be used to disentangle\nthe transverse from the longitudinal compression.\n\nAs before, almost all the detectors (except Trans1\n4) were plastic scintilla-\n\ntors with a groove. Inside the groove, a wavelength-shifting fiber was glued\n(see Fig. 110). The detectors were wrapped in several layers of Teflon tape\nand placed inside a copper collimator, as shown in Fig. 111. The copper\ncollimation increases the spatial resolution of the detectors by absorbing the\npositrons originating far from the scintillator position. The collimator was\nmounted above the target as shown in Fig. 113.\n\nFor each detector, we recorded the number of counts as a function of the\ntime difference between the hit in the detector and the time t = 0, given by\nthe muon passing through the entrance detector. In such a way, we obtained\ntime-spectra from which we can get the information about the muon motion.\n\n4 Trans1 was 1.5× 1.5 mm2 square scintillating fiber.\n\n\n\n158 mixed transverse-longitudinal compression\n\nFigure 107: Sketch of the detectors used in 2017 beam-time. The detectors Trans1-3\nwere used to monitor transverse compression of the muon beam, while\nthe detectors Tiles1-6 were monitoring the longitudinal compression.\nThe detectors Trans2 and Trans3 had dimensions 7× 5× 47 mm3, while\nTiles 1-6 had dimensions 10× 5× 7 mm3. Trans1 was a 1.5× 1.5 mm2\n\nsquare scintillating fiber.\n\nFigure 108: Detection efficiency of the detectors Trans1-3 projected in the yx-plane.\nThese detectors are sensitive to the transverse compression and drift in\nthe x-direction of the muon beam. The colors represent the probability\nto detect the positron from the muon decay as a function of the muon\ndecay position.\n\nFigure 109: Detection efficiency projected in the zx-plane for the three tile pairs in\ncoincidence: upstream (Tiles1 & Tiles4), middle (Tiles2 & Tiles5) and\ndownstream (Tiles3 & Tiles6). These detectors are sensitive to the lon-\ngitudinal compression of the muon beam, but also to the muon drift in\nx-direction. The colors represent the probability to detect the positron\nfrom the muon decay as a function of the muon decay position.\n\n\n\n6.3 measurements and results of the 2017 beam-time 159\n\nFigure 110: Picture of the plastic scintillators with wavelength-shifting fibers (glued\ninside the groove of the scintillator). The wavelength-shifting fibers are\nused to transport the scintillation light from the cold part of the setup\n(close to the target) to room temperature, where it is read-out by SiPMs.\n\nFigure 111: Picture of the plastic scintillators Tiles1-6 wrapped with Teflon tape and\nmounted inside the copper collimator.\n\nThe time-spectra of the Trans1-3 detectors are shown in Fig. 114. The time-\nspectra of the detectors Trans3 and Trans2 show first an increase, then a\ndecrease of the number of counts with time. This indicates that muons\npassed through the acceptance region of those detectors, i. e., they “flew-by\"\nthem. The counts in Trans1 are increasing with time, especially at late times,\nmeaning the muons were moving towards the tip of the target. Note that a\nlarge flat (in time) background limited the sensitivity of the Trans1 detector.\nThis background was most likely caused by the fiber extending upstream of\nthe target (until the muon beam entrance aperture, see Fig. 113), where there\nwas no collimation to prevent positron hits originating from muons stopped\noutside the target.\n\nThe time-spectra of the tile detectors in coincidence are shown in Fig. 115.\nWe can see that the number of counts in the middle tile pair increases at late\ntimes. In the upstream and downstream detectors the increase of counts is\n\n\n\n160 mixed transverse-longitudinal compression\n\nFigure 112: Picture of the target used in the 2017 beam-time.\n\nFigure 113: Picture of the target mounted on the cold finger. The copper collimator\nwith the various scintillators is placed above the target.\n\nmuch smaller, indicating that most of the muons were attracted to the center\nat z = 0 at late times. The reason for this small (but non-negligible) number\nof counts at late times in the upstream and downstream tiles is that the ac-\nceptance region of these detectors extends also outside the active region. In\nthis region we still have the transverse electric field (but no z-component) so\n\n\n\n6.3 measurements and results of the 2017 beam-time 161\n\nFigure 114: Measured time-spectra of the detectors Trans1-3 (see Fig. 107), corre-\nsponding to the conditions of Fig. 105.\n\n\n\n162 mixed transverse-longitudinal compression\n\nthat muons stopped there still drift in the x-direction (towards the detectors),\ncausing some hits in the upstream and downstream detectors.\n\nFigure 115: Measured time-spectra of the upstream, middle and downstream tile\npairs in coincidence (see Fig. 107), corresponding to the conditions of\nFig. 105.\n\nBy qualitatively comparing the measured time-spectra with the Geant4\nsimulation of the muon trajectories (Fig. 105), we can notice that the mea-\nsured muon compression and drift are slower than predicted by the simula-\ntions. For example, the time-spectra of the tile detectors indicate that muons\nwere still moving at very late times (10 µs), while according to the Geant4\nsimulations the muons should have reached the tip of the target within 8 µs.\n\nSince we previously observed a very good agreement between simulations\nand measurements (in the transverse and longitudinal compression test),\nthe most likely cause of these deviations is imperfect experimental setup.\nPossible experimental issues producing this disagreement include:\n\n• A misalignment between the target and magnetic field axes, shifting\nthe initial position of the muon beam inside the target. For example, if\nthe beam was injected in the bottom corner of the target, i. e., shifted in\n−x and −y direction, the muons would have to travel through a region\nof higher gas density and over a larger distance, thus increasing the\ncompression times. Such a misalignment was already observed in the\ntransverse compression measurements of 2015 beam-time. The smaller\nheight of the target used in 2017 beam-time makes the effects of the\nmisalignment more significant.\n\n• Deviations of the electric field strength and shape in the tip of the tar-\nget, caused by fairly limited precision of the target construction. This\neffect might be especially pronounced for such a small target height.\nThis could explain why the increase of the counts is very small in the\ntime-spectrum of the Trans1 detector (Fig. 114): if the deviation of the\n\n\n\n6.4 conclusions 163\n\nelectric field near the tip makes the muons crash into the target walls\ntoo soon, the muons would never reach the region near the tip of the\ntarget, where Trans1 detection efficiency is the highest.\n\n• Deviations of the detector and collimator positions from the design\nvalues due to the cool-down process and allowances of the mechanical\nconstruction.\n\nIn principle, the above described issues could be included in the Geant4\nsimulations, however, not without a significant effort. The more practical\napproach would be to improve the limitations of the setup.\n\n6.4 conclusions\n\nThe preliminary measurement of the mixed compression stage presented in\nthis chapter shows qualitatively the simultaneous transverse and longitudi-\nnal muon beam compression to be feasible. The factors affecting the com-\npression efficiency (electrical breakdown) and precision of the measurement\n(target construction and setup uncertainties) are expected to be improved\nduring 2019, allowing better comparison with the simulations and quantifi-\ncation of the compression efficiency in the beam-time planned for the end of\n2019.\n\n\n\n\n\n7\nN E X T S T E P S\n\n7.1 introduction\n\nAs we have seen in the previous chapter, the mixed transverse-longitudinal\ncompression represents a simpler (compared to the original proposal [16])\napproach for an efficient muon beam compression. However, electrical break-\ndown problems prevented us from achieving the optimal mixed compres-\nsion efficiency (3% instead of ≈ 7%).\n\nOne possible way of improving the compression efficiency has been ex-\nplored during the 2017 beam-time, namely, modifying the target geometry\nto allow for a larger extension (in the x-direction), so that more time is avail-\nable for completing the longitudinal compression. However, the particularly\nbad high voltage performance of the actual target prevented us from improv-\ning the compression efficiency.\n\nIn this chapter we will explore a different modification of the target: adding\na small extension (“nose”) at the tip of the target, as sketched in Fig. 116.\nThis configuration would also allow for longer muon drift in the x-direction,\nthus extending the time available for completing the compression. Moreover,\nthe small size of the “nose” could allow us to apply locally stronger electric\nfields (than inside the target volume), resulting in a smaller muon beam size.\n\nSuch a small “nose” represents also a natural next step for connecting the\nmixed transverse-longitudinal target to the next stages: the extraction into\nvacuum and re-acceleration stages. To extract the muons into the vacuum,\nwe need to be able to guide them through a small orifice with an area of\nabout 1 × 1 mm2 or smaller. This requires compressing the muon beam\nto as small size as possible, so that the muons fit through the orifice. Be-\nside efficient compression, also the muon position in the yz-plane needs to\nbe controlled precisely, so that the beam is centered at the orifice. In this\nway, the muon losses at the extraction orifice are minimized. Both of those\nrequirements are easier to achieve with the “nose” geometry (two parallel\nplates with a small gap): the ability to define a larger density gradient and\nstronger electric fields in the “nose” would make the compression more effi-\ncient. The muon beam position in the “nose” can be fine-tuned by adjusting\nthe temperature gradient and the electric field independently from the target\nregion.\n\n165\n\n\n\n166 next steps\n\nFigure 116: Sketch of the mixed transverse-longitudinal compression stage, that has\nbeen extended with a “nose”. Such a “nose” could help to further im-\nprove the compression efficiency of the mixed transverse-longitudinal\ncompression stage, and to precisely guide the muons through the ex-\ntraction orifice. After the muons exit the orifice, they need to be steered\ntowards a region of “good” vacuum and then accelerated along the −z-\ndirection in order to be extracted out of the 5 tesla magnetic field (as de-\npicted by the orange trajectory in the sketch). The sketch also shows the\npossible realization of the electrodes needed to guide the muon through\nthe full beam-line. For simplicity, in the extraction region, only the elec-\ntrodes on the bottom wall are drawn (the electrodes on the top have\nexactly the same shape).\n\nThe “nose” could be realized in two different ways:\n\n• “Cold nose”: where we keep the vertical density gradient at cryogenic\ntemperature throughout all the “nose”,\n\n• “Warm nose”: where the mixed transverse-longitudinal target is at\ncryogenic temperatures, as before, while the tip of the “nose” is warmed\nup to room temperature.\n\nWhich of these two options we will implement in the final setup will\ndepend on the exact design of the extraction stage. The “cold nose” would\nbe slightly simpler to realize, because there would be no transition from\ncryogenic to room temperatures. However, because of its higher gas density,\nthe vacuum pressure of the extraction and re-acceleration stage could be\nhigher than with the “warm nose”.\n\nIn each case (cold/warm) we need different electric fields. For the “cold\nnose” case, the electric field shape in the “nose” should be similar to the\nelectric fields inside the mixed compression target. For the “warm nose”\ncase, the gas density in the “nose” decreases with increasing x, so that the\nmuon drift direction rotates towards the Ê× B̂-direction. Hence, to guide\nthe muons in the +x-direction, we need to rotate the electric field until the\nx-component of the field vanishes. The resulting field is quite similar to the\nfields of the longitudinal stage with Ê× B̂ drift (see Chapter 5).\n\n\n\n7.2 target with cold extension 167\n\nThe purpose of the simulations presented in this chapter is to demonstrate\nthe feasibility of guiding the muons either through a cold or through a warm\n“nose”, while ideally improving the compression efficiency compared to only\nthe mixed compression target (without the “nose”). Only the most straight-\nforward implementations of the electric fields and density gradients in the\n“noses” will be considered in this chapter.\n\n7.2 target with cold extension\n\nA possible realization of a target with mixed compression and “cold nose”\nis shown in Figs. 117 and 119. The mixed longitudinal-transverse target of\nFig. 96 has been extended by elongating the bottom sapphire plate of the\nmixed compression target and by adding a second parallel top sapphire\nplate at a height of 4 mm from the bottom plate. As the two top sapphire\nplates are thermally decoupled from each other, it is possible to fine-tune\ntheir temperatures independently for better control of the muon motion.\n\nInitially, we keep the temperature of the bottom sapphire at 6.0 K and of\nboth top plates at 18.6 K (at 8.5 mbar), to have similar density gradients as\nin the mixed compression target (without “nose”). The temperature distri-\nbution in the target and “nose” obtained in this way is shown in Fig. 117.\n\nFigure 117: COMSOL Multiphysics® simulation of the temperature distribution in\nthe mixed compression target followed by a “cold nose”. The tempera-\nture at the bottom of the target and “nose” is 6.0 K and is defined by a\nsingle sapphire plate, while the temperature at the top walls is assumed\nto be 18.6 K and is defined by two separate sapphire plates.\nThe two coordinate systems used throughout this chapter are also\nshown in the sketch: the xyz coordinate system (where the xz plane\ncoincides with the target mid-plane), and the x ′y ′z ′ coordinate sys-\ntem, which is obtained by rotating the xyz system around the z-axis\nby θ = 15°, so that the x ′z ′ plane is parallel to the top and bottom walls\nof the “nose”.\n\nNote that, in this layout, the central planes of the mixed compression\ntarget and the “nose” are at an angle of θ = 15°. Such a configuration allows\n\n\n\n168 next steps\n\nus to use only a single bottom sapphire plate, thus simplifying the thermal\ncoupling of the target to the cold finger.\n\nTo simplify the description and calculation of the electric fields in the\ndifferent regions of the target-“nose” system we introduce two coordinate\nsystems, as shown in Fig. 117: the xyz system, which is the same as the\ncoordinate system used in previous chapter (where the xz plane coincides\nwith the target mid-plane), and the x ′y ′z ′ coordinate system, which is the\nxyz system rotated by 15° around the z-axis, so that the x ′z ′ plane is parallel\nto the top and bottom walls of the “nose” region.\n\nIn the mixed compression target, the electric field required for efficient\ntransverse compression is at 45° with respect to the x-axis, so that Ex = Ey,\nwhile the electric field required for the longitudinal compression is parallel\nto the z-axis and points towards z = 0. Hence, the electric field in the mixed\ncompression target region can be written as:\n\nE(x,y, z) =\n\n\nEx\n\nEy\n\n−Ez\nz\n|z|\n\n =\n\n\nEx\n\nEx\n\n−Ez\nz\n|z|\n\n , (229)\n\nwhere Ex and Ez are constants describing the strength of the different\ncomponents of the electric field in the xyz coordinate system.\n\nSuch a field can be transformed to the x ′y ′z ′ coordinate system by a sim-\nple rotation around the z-axis:\n\nE(x ′,y ′, z ′) =\n\n cos θ sin θ 0\n\n− sin θ cos θ 0\n\n0 0 1\n\n\n\n\nEx\n\nEx\n\n−Ez\nz\n|z|\n\n (230)\n\n=\n\n\n1.22Ex\n0.71Ex\n−Ez\n\nz\n|z|\n\n , (231)\n\nwhere we assumed θ = 15°. The z-component of the electric field is unaf-\nfected by the rotation.\n\nIn the “nose” the electric field is at 45° with respect to the x ′ axis, thus it\ntakes form:\n\nE(x ′,y ′, z ′) =\n\n\nEx\n\nEx\n\n−Ez\nz\n|z|\n\n . (232)\n\nWe can notice that the electric field has to be rotated at the transition\nbetween the target region and the “nose” region, from a 30° angle with\nrespect to the x ′-axis (see Eq. 231) to a 45° angle.\n\nWith such electric fields and a vertical density gradient, the muons should\nfirst drift in the x-direction through the target while simultaneously com-\npressing in the y- and z-direction. In the “nose” region the muons should\n\n\n\n7.2 target with cold extension 169\n\nthen drift in the x ′-direction, while still compressing further in the y ′- and\nthe z-direction.\n\nThe electrodes that produce the needed electric fields are shown in Fig. 118\n\n(unfolded Kapton) and in Fig. 119 (folded Kapton). If we apply the voltages\nto some of the electrodes as indicated in Fig. 118 and distribute it among the\nvarious lines via voltage dividers, we get the potential distribution on the\nelectrodes and on the target walls as depicted by the color scale in Fig. 119.\nUsing such potential on the walls as boundary condition, we can compute\nthe electric field and the electric potential within the target using a finite-\nelements method, as shown in Fig. 120.\n\nWith the HV given in Fig. 118, the average electric field strength within\nthe mixed compression target volume is Ex = 0.21 kV/cm, Ey = 0.18 kV/cm\n(E ′x = 0.24 kV/cm, E ′y = 0.12 kV/cm) and Ez = 0.05 kV/cm. In the “nose”\nE ′x = 0.16 kV/cm, E ′y = 0.14 kV/cm and Ez = 0.05 kV/cm.\n\nFigure 118: Lining of the Kapton foil needed for the mixed compression target fol-\nlowed by a“cold nose”. This design was obtained by extending the elec-\ntrodes of Fig. 95 to create the needed field in the “nose”. To obtain the\ndesired electric field, we apply voltage to some selected electrodes, as\nmarked in the sketch with arrows. On the other electrodes the potential\nis distributed via voltage dividers.\n\nAs a next step, we insert the density and the electric field maps simulated\nwith COMSOL Multiphysics® into the Geant4 simulation and we compute\nthe muon trajectories. In the simulation of Fig. 121, muons of 1 eV energy\nstart at x = −16 mm (2 mm from the side wall) with 18 mm spread in the\ny-direction and 40 mm spread in the z-direction. As visible from Fig. 121,\nwithin 2-5 µs, muons reach the x ′ = 20 mm plane (the beginning of the\n“nose”). At this point, their extension is reduced to about 0.54 mm in the y ′-\ndirection and to 0.8mm in the z-direction. The simulated muon distributions\nat x ′ = 20 mm are shown in Fig. 130 at the end of this chapter.\n\nThe probability that the muons reach the plane at x ′ = 20mm is 90% (with-\nout muon decay) and 20% (with muon decay). At this point, the longitudinal\n\n\n\n170 next steps\n\nFigure 119: Electrodes at the target walls obtained by folding the lined Kapton foil\nof Fig. 118. The color scale represents the potentials at the target walls\n(from COMSOL Multiphysics® simulation). These potentials are used\nas a boundary conditions for the simulation of the electric field inside\nthe target volume. They are obtained by mapping the potentials calcu-\nlated in a 2D simulation of the unfolded Kapton of Fig. 118 to the walls\nof the target.\n\ncompression is not fully completed, so that within an area of 1× 1 mm2 at\nx ′ = 20 mm there are 28% of the initial muons (without muon decay) and\n6.5% (with muon decay). Allowing more time for the compression in the\n“nose” improves the compression efficiency. The maximum compression ef-\nficiency within 1× 1 mm2 area and without muon decay is achieved around\nx ′ = 35 mm (see Fig. 131). If we include muon decay into the efficiency\nevaluation, the maximum efficiency of 8% is achieved around x ′ = 25 mm.\nIncreasing the nose length beyond x ′ = 25mm does not increase the number\nof muons potentially being extracted from an 1× 1 mm2 orifice, as the im-\nproved compression (smaller beam size) is outweighed by the muon decay\nlosses. Various efficiencies ε, ετ, ε1×1, ε1×1,τ\n\n1 and the beam spread in the y-\nand z-direction are summarized in the Table 5 at the end of this chapter.\n\nFrom the table we can see that the beam size (∆y, ∆z) does not improve\nsignificantly in the “nose”; in fact, the beam spread in the y- and z-direction\neven increases for x ′ > 30 mm. The increase of the muon spread in the y ′-\ndirection in the “cold nose” can be attributed to the relatively large “nose”\nheight (4 mm), which implies only moderate vertical density gradient. Fur-\nthermore, in this case, the muons drift close to the top sapphire plate of the\n“nose”, i.e., through a layer of relatively low gas density, which increases the\naverage muon energy, resulting in a larger spread in the z-direction.\n\n1 For the definition of these quantities see page 153.\n\n\n\n7.2 target with cold extension 171\n\nFigure 120: Simulated electric field lines (black lines) and electric potential (color)\nin the yx-plane at z = 10 mm (top), zx-plane at y = 0, i. e., target\nmid-plane (bottom left) and z ′x ′-plane at y ′ = −2mm, i. e., “nose” mid-\nplane (bottom right). For comparison, the “ideal” field lines obeying the\nrelation Ex = Ey = 3 Ez, which was the starting point for the electrode\ndesign, is represented by the gray lines. In the simulation, the potentials\nat the target walls of Fig. 119 were used as boundary condition.\n\nTo understand the relation between the beam size and the average muon\nkinetic energy, consider the evolution of the average muon energy versus the\nx ′-position, shown in Fig. 122 (dark blue points). The muons start with 1 eV\nenergy (around x ′ = −20 mm). After a drift of few mm in the x ′-direction\ntheir energy is increased to about 10 eV. This “drift” energy depends on the\ngas density and the strength of the applied electric field. At the transition be-\ntween the mixed compression target and the “cold nose” the average muon\nenergy suddenly drops to only few eV. This drop in energy is caused by the\ngradual change of the electric field angle at the transition between target and\nthe “nose”, from 30 ° to 45 ° with respect to the x ′-axis. Since this change is\nnot perfectly matched to the change of gas density in this region, the muons\ndrift slightly downwards, into a region of higher gas density so that their\n\n\n\n172 next steps\n\nFigure 121: Projections of the muon trajectories in the y ′x ′-plane (top) and in the\nzx ′-plane (bottom). The trajectories were simulated in Geant4 by using\nthe simulated electric field of Fig. 120 (with maximum HV = 4.8 kV)\nand the simulated density gradient of Fig. 117 (6.0-18.6 K at 8.5 mbar)\nand 5 tesla magnetic field.\nThe 1 eV muons start at x = −16 mm with 18 mm spread in the y-,\nand 40 mm spread in the z-direction. Within ≈ 2− 5 µs they reach the\nbeginning of the “nose” at the x ′ ≈ 20 mm plane, where their spread is\n0.54 mm in the y-, and 0.8 mm in the z-direction. After that, the muons\ndrift mostly in x ′-direction through the “nose” with drift velocity of\n≈ 10 mm µs−1.\n\n\n\n7.2 target with cold extension 173\n\naverage energy is reduced. This decrease of the average muon energy leads\nto a decrease of the muon spread in the y- and z-direction. After passing\nthe transition region, the muons drift slightly upwards, so that their average\nenergy increases again, along with the beam size in the y- and z-direction.\n\nFigure 122: Evolution of the average muon swarm energy as a function of the x ′-\nposition, for the three “nose” configurations investigated in this chapter:\nthe “cold nose” with temperature gradient from 6.0 to 18.6 K (extracted\nfrom the simulation of Fig. 121), the “cold nose” with temperature gra-\ndient from 6.0 to 25.0 K (extracted from the simulation of Fig. 124), and\nthe “warm nose” (extracted from the simulation of Fig. 129). In all three\ncases, the muons initially have 1 eV energy. During the drift and com-\npression inside the target their energy increases to about 10 eV. At the\ntransition from the target to the “nose” (at x ′ ≈ 20 mm) their energy\ndecreases temporarily. This is because the shape of the electric field\naround this point makes them drift slightly downwards, into a region\nof higher gas density. After x ′ ≈ 30 mm their energy increases to dif-\nferent values depending on the density of the gas where the muons are\nmoving in the “nose”.\n\nBy comparing Figs. 121 and 122, we can see that the smallest beam size\nis obtained at x ′ = 20 mm where the muon average energy is the lowest. It\nwould be advantageous if we could keep such size throughout the “nose”.\nThis can be achieved by keeping the muons in the lower part of the “ cold\nnose”, where the He gas density is higher. One possible way to do this is\nby changing slightly the electric fields in the “nose”, either by rotating the\nelectric field so that its angle is smaller than 45 ° with respect to the x ′-axis\nor by increasing the strength of the electric field.\n\n\n\n174 next steps\n\nAlternatively, we can alter the density gradient strength in the “nose”,\ndN/dy ′, either by decreasing the “nose” height (while keeping the temper-\nature at top and bottom walls of the “nose” the same as before, i. e., 18.6 K\nand 6.0 K, respectively) or by increasing the temperature on the top sapphire\n(while keeping the same temperature at the bottom sapphire and the same\nnose height). In the following, we explore the second approach, since it al-\nlows us to use the same electric field and adjust only the density gradient, as\nshown in Fig. 123. This is also more advantageous from the practical point\nof view, because it allows us to control the beam size and position simply by\ncontrolling the top sapphire temperature, without having to exchange the\ntarget.\n\nFigure 123: Temperature distribution for a target with a “cold nose” having a\nstronger temperature gradient in the “nose”, simulated with COMSOL\nMultiphysics®. In the target region the bottom temperature is 6 K, while\nthe top temperature is 18.6 K. In the “nose” region the bottom tempera-\nture is 6 K, while the top temperature is 25 K.\n\nBy implementing the density gradient of Fig. 123 in combination with the\nelectric fields of Fig. 120 into the Geant4 simulation, we obtain the muon\ntrajectories of Fig. 124. The initial parameters of the muon beam are the\nsame as assumed for the simulation of Fig. 121. The muon trajectories inside\nthe mixed compression target are not altered compared to Fig. 121 since the\nelectric fields and the density gradient in the target region were not changed.\nHowever, inside the “nose”, the trajectories are different: the muons drift\nthrough the lower part of the “nose”, resulting in a smaller beam size. For\nthe “nose” with stronger density gradient, the FWHM of the muon beam at\nthe x ′ = 20mm plane is 0.45mm in the y ′-direction and 0.72mm z-direction\n(see Fig. 132).\n\nThe probability that muons reach the plane at x ′ = 20 mm is 90% without\nmuon decay and 20% including muon decay. If we consider only muons\nthat fit within an 1× 1 mm2 area, these probabilities are reduced to 32.6%\n(without muon decay) and 7.4% (including muon decay), which is an im-\nprovement by about 15% compared to the previous case, with weaker den-\nsity gradient in the “nose”. Allowing muons to propagate further through\n\n\n\n7.2 target with cold extension 175\n\nthe “nose” improves the efficiency ε1×1 even more, as visible in Table 6.\nAt around x ′ = 35 mm both longitudinal and transverse compression are\ncompleted. The muons y ′-, z-, arrival time- and energy-distribution at\nx ′ = 35 mm are shown in Fig. 133.\n\nFigure 124: Projections of the muon trajectories in the y ′x ′-plane (top) and in the\nzx ′-plane (bottom). The trajectories were simulated in Geant4 using the\nsimulated electric field of Fig. 120 (with maximum HV = 4.8 kV) and\nthe simulated temperature gradients of Fig. 123 (6-18.6 K in the target\nand 6-25 K in the “nose” at 8.5 mbar) and a 5 tesla magnetic field.\nThe trajectories inside the target are similar as in Fig. 121. However, the\nstronger temperature gradient in the “nose” (compared to the condi-\ntions underlying Fig. 121) keeps the muons traveling through the lower\npart of the “nose”, resulting in a smaller beam size.\n\nWe can conclude that by increasing the temperature gradient in the “nose”\nwe improve the beam quality: we achieve smaller average muon energy (see\nFig. 122), leading to smaller spread in the y ′- and z ′-directions and thus\nimproving the efficiency ε1×1. However, the decrease in the average muon\nenergy leads also to a slightly slower drift and so that the total efficiency\n\n\n\n176 next steps\n\nε1×1,τ (including muon decay) is not significantly improved: the increase in\nthe efficiency ε1×1 is counteracted by the increased muon losses through the\nmuon decay.\n\n7.3 target with warm extension\n\nWe now turn our attention to the “warm nose”. In this case, the end of the\n“nose” (at x ′ = 45 mm) is heated to 300 K, while the bottom and top walls\nof the mixed compression target are kept at 6.0 K and 18.6 K, respectively\n(see Fig. 125). By setting such boundary conditions we still have a vertical\ndensity gradient inside the mixed compression target, needed for the trans-\nverse compression, while in the “nose” the temperature will rapidly change\nfrom cryogenic values at the transition target-“nose” to room temperature at\nthe tip of the “nose”. This horizontal temperature gradient can be controlled\nroughly by varying the length of the bottom sapphire in the +x ′-direction,\nas shown in Fig. 125.\n\nIn order to steer the muons correctly through such a large range of gas\ndensities, the electric field direction and strength inside the “nose” need to\nbe adapted appropriately. When going from about 10 K to 300 K (at 8.5mbar\nof He), the muon drift angle with respect to the Ê× B̂ direction changes from\nabout 45° to almost 0° (see Chapter 3). Therefore, to keep muons drifting\nin the +x ′-direction, the electric field in the “nose” needs to be rotated from\n45° to 90° (with respect to the x ′-axis).\n\nThe simplest implementation of the required electric field can be realized\nwith the Kapton foil shown in Fig. 126, folded as shown in Fig. 127. In the\ntarget region the lining is the same as for the mixed compression target of\nFig. 118. In the “nose” region the electrodes are oriented parallel to the x ′-\ndirection (see Fig. 127). In this way, the x ′-component of the electric field in\nthe “nose” vanishes. A potential difference of 0.11 kV between the top and\nthe bottom lines in the “nose” creates a y ′-component of the electric field,\nthat ensures the muon drift towards the target tip (extraction orifice), while\nthe increase of the potential on the lines for increasing |z| creates an electric\nfield component in the z-direction, pointing towards z = 0, that ensures first\ncompression and then confinement in the z-direction.\n\nThe simulated electric field shape and the electric potential are shown in\nFig. 128, for the electrodes of Fig. 126, folded as in Fig. 127. Note that in\nthis simulation, the strength of the electric field in the “nose” was increased\nslightly above the design value by applying a larger voltage at the bottom\ncentral (z = 0) electrode of the “nose” (110 V instead of 70 V, as indicated\nin Fig. 126). This does not significantly affect the electric fields inside the\ntarget. The average electric field strength inside the target volume is Ex =\n\n0.20 kV/cm, Ey = 0.19 kV/cm (E ′x = 0.24 kV/cm, E ′y = 0.13 kV/cm) and\nEz = 0.05 kV/cm and in the “nose” it is E ′x ≈ 0 kV/cm, E ′y = 0.25 kV/cm\nand Ez = 0.06 kV/cm when the HVs of Fig. 126 are applied.\n\nBy using the electric fields of Fig. 128 and the simulated densities of\nFig. 125 in the Geant4 simulation, the trajectories of Fig. 129 are obtained.\nAs before, muons of 1 eV energy start at x = −16 mm (2 mm from the\n\n\n\n7.3 target with warm extension 177\n\nFigure 125: COMSOL Multiphysics® simulation of the temperature gradient of a\ntarget with “warm nose”. In the target the bottom and the top sapphires\nwere kept at 6 K and 18.6 K, respectively, and the “nose” end at 300 K.\nIn the “nose” the temperature thus increases along the x ′-direction from\nfew K to 300 K at the end of the “nose”.\n\nFigure 126: Electrodes on the unfolded Kapton foil for the configuration with\n“warm nose”. This design was obtained by extending the electrodes\nof Fig. 95 to create the needed field in the “nose” with no x ′-component.\nThe electrode density was increased compared to Fig. 95 to improve the\ndefinition of the electric field in the “nose”. To obtain the desired electric\nfield, we apply suited voltages to some selected electrodes, as marked\nin the sketch. On the remaining electrodes the potential is distributed\nuniformly via voltage dividers.\n\n\n\n178 next steps\n\nFigure 127: Electrodes at the target walls obtained by folding the lined Kapton foil\nof Fig. 126. The color scale represents the potentials at the target walls\n(from COMSOL Multiphysics® simulation). These potentials are used\nas a boundary conditions for the simulation of the electric field inside\nthe target volume (see Fig. 128). They are obtained by mapping the\npotentials from a 2D simulation of the unfolded Kapton of Fig. 126 to\nthe walls of the target.\n\nFigure 128: Simulated electric field lines (black lines) and electric potential (color) in\nthe yx-plane at z = 10mm (top), in the zx-plane at y = 0, i. e., target mid-\nplane (bottom left) and in the z ′x ′-plane at y ′ = −2 mm, i. e., “nose”\nmid-plane (bottom right). In the simulation the electrode voltages of\nFig. 127 were assumed.\n\n\n\n7.4 conclusions 179\n\nside wall) with 18 mm spread in the y-direction and 40 mm spread in the\nz-direction. The simulation confirms that it is possible to guide the muons\nfrom cryogenic temperatures to room temperature without excessive losses.\nHowever, we also see that after about 15 mm drift in the “warm nose” some\nmuons hit the top wall. Indeed, even though the gas density in the “nose” is\nlow, the drift velocity still has a non-vanishing component in the E-direction,\nwhich causes a slow drift of the muons in the y ′-direction. This drift in\nthe y ′-direction eventually limits the length of the “nose” for a given “nose”\nheight.\n\nThe efficiencies and the beam size at several positions along the “nose”\nare summarized in Table 7. From this table we can see that the beam size\nincreases as the muons propagate through the “nose”. At x ′ = 20 mm, the\nmuon beam extension (FWHM) is about 0.36 mm in the y ′-direction and\n1.60 mm in the z-direction (see distributions of Fig. 134). At x ′ = 35 mm, the\nbeam spread is increased to 0.63 mm in the y ′-direction and 1.68 mm in the\nz-direction (see Fig. 135).\n\nIn the y ′-direction (transverse direction) the increase of the muon swarm\nspread during the drift in the x ′-direction is caused by the lack of a vertical\ndensity gradient that would counteract the muon swarm diffusion in y ′-\ndirection. In the z-direction the muon spread is larger than in the “cold\nnose” because the average muon energy is higher at such low gas density\n(see red points in Fig. 122). Indeed, the muons effectively oscillate in the\nz-direction (around z = 0) with an amplitude that increases with energy.\nThese increased spreads are also reflected in the values of the efficiency ε1×1,\nwhich is about a factor of 2 smaller than in the “cold nose”. However, due the\nfaster muon drift, the losses through the muon decay are reduced compared\nto the “cold nose” case, so that the efficiencies ε1×1,τ for the “warm nose”\nand the “cold nose” are almost the same.\n\nWe can conclude that the guiding of muons through a low gas density\nafter having completed transverse and longitudinal compression is feasible,\neven with relatively simple choices of electric fields and density gradients.\nIt is however advantageous the keep the length of the “nose” as short as\npossible to avoid muon losses and decrease of beam quality.\n\n7.4 conclusions\n\nWe have shown that it is possible to couple and efficiently transport a muon\nbeam that has been (partially) compressed in a mixed compression target\ninto a small channel (“nose”). The simulations show that the compression\nefficiency can be further improved compared to only a mixed compression\ntarget by adding such a “nose”. Both “cold nose” and “warm nose”, i.e, a\n“nose” with cryogenic vertical temperature gradient or horizontal tempera-\nture gradient from cryogenic to room temperature were shown to be suitable\nas an extension to the mixed compression target. The Figs. 132-135 as well\nas Tables 5-7 summarize the beam sizes and the various compression effi-\nciencies for three different “nose” configurations. The “cold nose” appears\nto be slightly superior because it allows more efficient transverse and longi-\n\n\n\n180 next steps\n\nFigure 129: Projections of the muon trajectories in the y ′x ′-plane (top) and in the\nz ′x ′-plane (bottom). The trajectories were simulated in Geant4 by using\nthe simulated electric field of Fig. 128 (with maximum HV = 4.8 kV) and\nthe simulated temperature gradient of Fig. 125 (6-18.6 K at 8.5 mbar in\nthe target, increasing to 300 K in the “nose”) and a 5 tesla magnetic field.\nThe 1 eV muons start around x = −16 mm with 18 mm spread in the\ny-, and 40 mm spread in the z-direction. Within about 2 − 5 µs they\nreach the beginning of the “nose” at x ′ ≈ 20 mm, where their spread\n(FWHM) is 0.36 mm in the y-, and 2.1 mm in the z-direction. After that,\nthe muons drift mostly in x ′-direction through the “nose” with drift\nvelocity of about 30 mm µs−1. The beam size increases slightly while\nmuons propagate in the x ′-direction because of the absence of a vertical\ntemperature gradient.\n\ntudinal compression, yielding a smaller muon beam at lower kinetic energy.\nHowever, if we consider also the muon losses from the muon decay, the\ndifferences are less significant, because the losses due to less efficient com-\npression with the “warm nose” are compensated by an about 3 times faster\ndrift velocity.\n\n\n\n7.4 conclusions 181\n\nEven though the “cold nose” and the “warm nose” show very similar\nfinal compression efficiencies, ε1×1,τ, it would be interesting to test both\nconfigurations experimentally, as explained in the following.\n\nThe “cold nose” has the advantage of being fairly simple to implement\nand it could possibly allow us to improve the quantification of the compres-\nsion efficiency of the mixed compression stage experimentally. Indeed, the\nmixed compression efficiency (only in the target region) was extracted in-\ndirectly by comparing the measured time-spectra with Geant4 simulations,\nas shown in Chapter 6. Adding a horizontal “nose” with an 1 × 1 mm2\n\norifice and measuring the transmitted number of muons would allow us\na more precise quantification of the achieved mixed compression efficiency\nand beam quality.\n\nOn the other hand, the “warm nose” is similar in density to the extraction\nstage (low density, almost pure Ê× B̂ drift), so it would allows us to test the\nmuon beam propagation at such conditions. Therefore, even if we do not\nuse it in the final beam-line design, it might be useful as a preliminary study\nof the muon beam manipulation in the vacuum extraction stage.\n\nIn both cases, it was observed that the final beam quality is quite sensitive\nto the electric field strength and gas density gradient, which is most likely\nrelated to the average muon energy. This should be studied more systemati-\ncally, for various electric field strengths and shapes (in particular for various\nEx/Ez ratios) and for various gas density gradients.\n\n\n\n182 next steps\n\nFigure 130: Muon z-distribution (top left), y ′-distribution (top right), arrival time\n(bottom left) and energy (bottom right) distribution at x ′ = 20 mm for\nthe“cold nose” configuration. These distributions were extracted from\nthe simulation of Fig. 121. The shaded regions show the fraction of\nmuons within ±0.5 mm interval centered at the mean of the z- and the\ny-distributions. About 31% (neglecting the losses due to muon decay)\nof the muons are within an 1× 1 mm2 area.\n\nFigure 131: Similar to Fig. 130, but at x ′ = 35 mm. In this case 61% (neglecting\nthe losses due to muon decay) of the muons are within an 1× 1 mm2\n\narea. Even though the FWHM in the z-direction became larger, the tails\nof the distribution are reduced, yielding a better efficiency compared to\nFig. 130.\n\n\n\n7.4 conclusions 183\n\nFigure 132: The muon z-distribution (top left), y ′-distribution (top right), arrival\ntime (bottom left) and energy (bottom right) distribution at x ′ = 20 mm\nfor the “cold nose” configuration with a stronger temperature gradient.\nThese distributions were extracted from the simulation of Fig. 124. The\nshaded regions show the fraction of muons within ±0.5 mm interval\ncentered at the mean of z- and y ′-distributions. About 37% (neglecting\nthe losses due to muon decay) of the muons are within an 1× 1 mm2\n\narea.\n\nFigure 133: Similar to Fig. 132, but at x ′ = 35 mm. About 67% (neglecting the losses\ndue to muon decay) of the muons are within an 1× 1 mm2 area.\n\n\n\n184 next steps\n\nFigure 134: The muon z-distribution (top left), y ′-distribution (top right), arrival\ntime (bottom left) and energy (bottom right) distribution at x ′ = 20 mm\nfor the “warm nose” configuration. These distributions were extracted\nfrom the simulation of Fig. 129. The shaded regions show the fraction\nof muons within ±0.5 mm interval centered at the mean of z- and y ′-\ndistributions. About 29% (neglecting the losses due to muon decay) of\nthe muons are within 1× 1 mm2 area.\n\nFigure 135: Similar to Fig. 134, but at x ′ = 35 mm. Only 29% (neglecting the losses\ndue to muon decay) of the muons are within an 1× 1 mm2 area.\n\n\n\n7.4 conclusions 185\n\nTable 5: “Cold nose”\n\nx’ ∆y ′ ∆z tcomp\nε ετ ε1×1 ε1×1,τ\n\n(mm) (mm) (mm) (µs)\n\n20 0.54 0.80 3.5 89.7% 19.8% 28.1% 6.5%\n\n25 0.49 0.88 4.2 89.5% 14.9% 46.9% 8.0%\n\n30 0.41 1.12 4.7 89.1% 11.8% 51.0% 6.9%\n\n35 0.45 1.12 5.1 88.5% 9.5% 53.9% 5.9%\n\n40 0.50 1.20 5.6 87.7% 7.8% 52.5% 4.7%\n\n45 0.40 1.28 6.0 86.9% 6.4% 51.5% 3.8%\n\nTable 6: “Cold nose” with a strong temperature gradient\n\nx’ ∆y ′ ∆z tcomp\nε ετ ε1×1 ε1×1,τ\n\n(mm) (mm) (mm) (µs)\n\n20 0.45 0.72 3.5 89.1% 20.0% 32.6% 7.4%\n\n25 0.32 0.80 4.1 88.9% 15.1% 48.6% 8.3%\n\n30 0.36 0.88 4.7 88.6% 11.7% 57.6% 7.8%\n\n35 0.31 0.96 5.2 88.3% 9.2% 59.5% 6.3%\n\n40 0.31 0.96 5.7 87.9% 7.3% 61.2% 5.1%\n\n45 0.32 0.80 6.2 87.6% 5.7% 67.8% 4.4%\n\nTable 7: “Warm nose”\n\nx’ ∆y ′ ∆z tcomp\nε ετ ε1×1 ε1×1,τ\n\n(mm) (mm) (mm) (µs)\n\n20 0.36 1.60 2.8 85.5% 25.4% 33.0% 9.3%\n\n25 0.45 1.92 3.0 84.0% 22.7% 32.3% 8.4%\n\n30 0.54 1.76 3.2 82.5% 20.5% 32.4% 7.7%\n\n35 0.63 1.68 3.4 81.0% 18.4% 31.6% 7.0%\n\n40 0.58 1.68 3.6 79.1% 16.5% 32.5% 6.5%\n\n45 1.40 1.52 3.8 76.3% 14.7% 32.1% 6.0%\n\nSummary of the numbers relevant for quantifying the compression efficiency at\nvarious x ′, for different gas density distributions in the “nose”. These numbers\nwere extracted from the Geant4 simulations of Figs. 121, 124 and 129 (with 104\n\nsimulated muons for each simulation). The quantities ∆y ′ and ∆z are FWHM of\nmuon y ′- and z- distributions at the various x ′. The time tcomp is the average\ncompression time (time it takes the muons to reach the position x ′). The efficiencies\nε, ετ, ε1×1 and ε1×1,τ are defined on page 153.\n\n\n\n\n\n8\nS U M M A RY A N D O U T L O O K\n\nThis thesis was concerned with the development of the muCool device, fea-\nturing a novel concept for the phase space compression of a positive muon\nbeam. In this compression scheme the muons are first stopped in a he-\nlium gas target and successively their motion is manipulated in a few stages\nusing complex electric and magnetic fields and density gradients to steer\nthe muons into one spot. From this spot the muons are extracted through\na small orifice from the gas target into vacuum, where re-acceleration oc-\ncurs. In the course of this thesis, the first two stages of this scheme, the\ntransverse and the longitudinal compression stages, have been successfully\ndemonstrated. The measurements were compared to Geant4 simulations, in\nwhich low-energy muon-He interactions (low-energy elastic collisions and\ncharge exchange) were implemented. Good agreement was observed, thus\nallowing us to validate the basic principle of the muCool device.\n\nBased on the successful demonstration of transverse and longitudinal com-\npression, it was possible to take the proposed idea [16] one step further\nand combine the demonstrated two stages into a single stage, the so-called\nmixed transverse-longitudinal compression stage. This new concept was\nfirst demonstrated using Geant4 simulations, and then experimentally veri-\nfied. However, the experimentally achieved compression efficiency was lim-\nited by several experimental problems, mainly electrical breakdown, limiting\nthe electric field strength. Still, provided that these issues are solved, a single\nmixed compression stage could be a viable simpler alternative to the original\nproposal presented in Ref. [16], with 3 compression stages. Moreover, this\nsetup would allow us to test the extraction of the low-energy muon beam\ninto vacuum in the near future.\n\nTo efficiently extract muons into the vacuum, it is necessary to guide them\nfrom the high gas density of the target, where the compression occurs, to\nthe low density region of the extraction (vacuum), while maintaining the\nsmall beam size. Several realistic schemes have been developed in this thesis,\nusing the previously gained insight into the muon drift in the He gas with\ndensity gradients, complex electric fields and strong magnetic field. The\nproof of principle of these schemes was verified using Geant4 simulations in\nconjunction with COMSOL Multiphysics® simulations of the electric fields\nand gas density distributions.\n\nGiven the overall good agreement between our simulations and experi-\nmental data, we can conclude that the full muCool phase space compression\nscheme with efficiency of 10−3, as predicted in [16], seems attainable. More-\nover, the theory underlying this compression scheme appears to describe\nwell our measurements within the current experimental sensitivity. Hence,\nthe Geant4 simulation based on this theory can be reliably used in future to\ndesign the next stages.\n\n187\n\n\n\n188 summary and outlook\n\nStill, it should be pointed out that our current theoretical model does\nnot include all possible muon-He interactions. Moreover, the ones that are\nincluded are based on scaled proton-He cross sections. In this thesis, the\nlimitations of such approximate scaling schemes have been discussed. If the\nexperimental sensitivity is improved in the future, further theoretical inves-\ntigation might be needed, concerning the sensitivity to the cross sections,\ndirect calculation of muon-He cross sections, better treatment of the energy\nlosses in charge exchange and possible molecular ion (Heµ+) formation.\n\n\n\nA P P E N D I X\n\n189\n\n\n\n\n\nA\nM O T I O N O F C H A R G E D PA RT I C L E S I N C R O S S E D\nE L E C T R I C A N D M A G N E T I C F I E L D S I N T H E VA C U U M\n\nIn this appendix, the derivation of the equation of motion of charged parti-\ncles in vacuum and in crossed electric and magnetic fields is given, following\nRef. [73]. We will assume in the following that the magnetic field is oriented\nalong the z-axis, B = (0, 0,B), while the electric field is parallel to the y-\naxis, E = (0,E, 0). The particle of positive charge +e and mass m is initially\nat the origin of the coordinate system (x = y = z = 0) and has a velocity\nv0 =\n\n(\nvx,0, vy,0, vz,0\n\n)\n. The Lorentz force acting on the particle is:\n\nF = e (E + v×B) , (233)\n\nor written separately for each component:\n\nFx = m\ndvx\n\ndt\n= evyB (234)\n\nFy = m\ndvy\n\ndt\n= eE− evxB (235)\n\nFz = m\ndvz\n\ndt\n= 0, (236)\n\nwhere vx, vy and vz are the x-, y- and z components of the particle velocity.\nFrom Eq. (236) we see that the z-component of the velocity is constant in\ntime, since there is no force in the z-direction acting on the particle:\n\nvz(t) = vz,0. (237)\n\nThe other velocity components can be obtained by solving the differential\nequations of Eqs. (234) and (235). If we differentiate Eq. (235) with respect to\ntime and combine it with Eq. (234) we obtain a differential equation for vy:\n\nd2vy\n\ndt2\n= −\n\n(\neB\n\nm\n\n)2\nvy = −ω2vy, (238)\n\nwhere we introduced the cyclotron frequency ω = eB\nm . The solution of this\n\ndifferential equation is of the form:\n\nvy(t) = A sinωt+B cosωt, (239)\n\nwhere A and B are constants to be determined from the initial conditions.\nThe first initial condition is:\n\nvy(t = 0) = vy,0, (240)\n\nwhich combined with Eq. (239) leads to B = vy,0. The second initial condi-\ntion can be obtained from Eq. (235):\n\ndvy\n\ndt\n(t = 0) =\n\neE\n\nm\n−\neB\n\nm\nvx,0. (241)\n\n191\n\n\n\n192 motion of charged particles in crossed electric and magnetic fields in the vacuum\n\nBy differentiating Eq. (239) and combining it with the second initial condi-\ntion of Eq. (241) we get the solution for the constant A:\n\nA =\nE\n\nB\n− vx,0. (242)\n\nSummarizing, the solution for the y-component of the velocity is:\n\nvy(t) =\n\n(\nE\n\nB\n− vx,0\n\n)\nsinωt+ vy,0 cosωt. (243)\n\nBy inserting Eq. (243) into Eq. (234) we obtain the differential equation for\nvx:\n\ndvx\n\ndt\n=\neB\n\nm\nvy = ω\n\n(\nE\n\nB\n− vx,0\n\n)\nsinωt+ωvy,0 cosωt. (244)\n\nThe solution of this equation can be obtained by integration with respect to\ntime:\n\nvx(t) − vx(t = 0) = ω\n\n(\nE\n\nB\n− vx,0\n\n) t∫\n0\n\nsinωt ′ dt ′ +ωvy,0\n\nt∫\n0\n\ncosωt ′ dt ′\n\n(245)\n\nvx(t) = vx,0 +\n\n(\nE\n\nB\n− vx,0\n\n)\n(1− cosωt) + vy,0 sinωt (246)\n\nvx(t) =\nE\n\nB\n(1− cosωt) + vx,0 cosωt+ vy,0 sinωt. (247)\n\nIn the special case where the particle is initially at rest (vx,0 = vy,0 =\n\nvz,0 = 0), the solutions for the velocity simplify to:\n\nvx(t) =\nE\n\nB\n(1− cosωt) (248)\n\nvy(t) =\nE\n\nB\nsinωt (249)\n\nvz(t) = 0. (250)\n\nIn this case, the displacements in the x-, y- and z-direction as a function\nof time are:\n\nx (t) =\nE\n\nB\n\n1\n\nω\n(ωt− sinωt) (251)\n\ny (t) =\nE\n\nB\n\n1\n\nω\n(1− cosωt) (252)\n\nz (t) = 0, (253)\n\ncalculated by integrating Eqs. (248), (249) and (250), respectively.\n\n\n\nB\nE L A S T I C C O L L I S I O N S\n\nIn this appendix, the expressions for the momentum and energy change of\nthe particle A (projectile) in an elastic collision with particle B (target) will\nbe derived. The variables used are defined in Chapter 2.\n\nchange of momentum in an elastic collision\n\nThe momentum change of the particle A in the laboratory frame is given by:\n\n∆pA = p ′A − pA (254)\n\n= mA\n(\nv ′A − vA\n\n)\n. (255)\n\nThe momentum change is the same in the center-of-mass frame, as will be\nshown in the following. The relation between velocities in laboratory and\ncenter-of-mass frame is:\n\nvA = vA,CM + vC (256)\n\nv ′A = v ′A,CM + v ′C, (257)\n\nwhere vC and v ′C is the velocity of center of mass before and after the colli-\nsion, respectively and vA,CM and v ′A,CM is the velocity of particle A in the\ncenter-of-mass frame before and after the collision. We plug this into Eq.\n(255):\n\n∆pA = mA\n(\nv ′A,CM + v ′C − vA,CM − vC\n\n)\n(258)\n\n= mA\n(\nv ′A,CM − vA,CM\n\n)\n(259)\n\n= p ′A,CM − pA,CM (260)\n\n= ∆pA,CM, (261)\n\nwhere we used the fact that the velocity of center of mass does not change\nin the collision: vC = v ′C.\n\nThe momentum in the center-of-mass frame can be written in terms of the\nrelative particle velocity vr = vA − vB:\n\npA,CM = mrvr, (262)\n\nwhere mr = mAmB\n\nmA+mB\nis the reduced mass. Thus, Eq. (261) becomes:\n\n∆pA = mr\n(\nv ′r − vr\n\n)\n. (263)\n\nWe can separate ∆pA into components parallel and perpendicular to vr (see\nFig. 136):∣∣∣(∆pA)‖\n\n∣∣∣ = mr ∣∣v ′r∣∣ cosχ−mr |vr| (264)\n\n= mr |vr| (1− cosχ) (265)\n\n193\n\n\n\n194 elastic collisions\n\n∣∣(∆pA)⊥\n∣∣ = mr ∣∣v ′r∣∣ sinχ− 0 (266)\n\n= mr |vr| sinχ, (267)\n\nwhere we noted that the relative velocity only changes the direction (by an\nangle χ) in the elastic collision, while the norm is conserved: |v ′r| = |vr|.\n\nFigure 136: Change of the relative velocity between the two particles after an elastic\ncollision.\n\nIf the potential governing the elastic collision between particles A and\nB is spherically symmetric (i. e., it depends only on the relative distance r\nbetween the particles), over many collisions the momentum change perpen-\ndicular to the relative velocity will average out to zero:∣∣∣(∆pA)⊥\n\n∣∣∣ = mr |vr| sinχ = 0. (268)\n\nThus, we are left only with the momentum change parallel to the relative\nvelocity:\n\n∆pA = mrvr(1− cosχ). (269)\n\nchange of energy in an elastic collision\n\nThe change of energy of particle A in the laboratory frame is given by:\n\n∆EA =\n1\n\n2\nmAv ′2A −\n\n1\n\n2\nmAv2A (270)\n\n=\n1\n\n2\nmA\n\n(\nv ′A,CM + v ′C\n\n)2\n−\n1\n\n2\nmA (vA,CM + vC)\n\n2 (271)\n\n=\n1\n\n2\nmA\n\n(\nv ′2A,CM − v2A,CM + v ′2C − v2C + 2v ′A,CM · v ′C − 2vA,CM · vC\n\n)\n.\n\nSince the velocity of center of mass does not change in the collision: vC = v ′C,\nand the particle velocities in the center-of-mass frame only change direction,\nbut not the norm: |v ′A,CM| = |vA,CM|, the energy change simplifies to:\n\n∆EA = mAvC ·\n(\nv ′A,CM − vA,CM\n\n)\n(272)\n\n= vC ·∆pA,CM (273)\n\n= vC ·∆pA. (274)\n\n\n\nelastic collisions 195\n\nIf the second particle is initially at rest (vB = 0), the velocity of the center of\nmass vC is parallel to the relative velocity vr:\n\nvC =\nmAvA +mBvB\nmA +mB\n\n=\nmA\n\nmA +mB\nvr, (275)\n\nso that the energy change of particle A becomes:\n\n∆EA = vC ·∆pA = vC\n\n∣∣∣(∆pA)‖\n∣∣∣ , (276)\n\nwhere the momentum change component (∆pA)‖ parallel to vr is given by\nEq. (265). It follows that the energy change in the laboratory reference frame\nis:\n\n∆EA =\nmA\n\nmA +mB\nmrv\n\n2\nr(1− cosχ) (277)\n\n= 2\nmA\n\nmA +mB\nECM(1− cosχ), (278)\n\nwhere χ is the scattering angle of the projectile in the center-of-mass system\nand ECM = 1\n\n2mrv\n2\nr .\n\n\n\n\n\nC\nE L A S T I C S C AT T E R I N G I N Q U A N T U M M E C H A N I C S\n\nIn this appendix, a short outline of the treatment of the elastic scattering in\nquantum mechanics will be given, following Refs. [34, 41, 43].\n\nIn quantum mechanics, the starting point for determining the cross sec-\ntions is the time-independent Schrödinger equation (instead of the classical\nenergy conservation of Eq. (60)):[\n\n−\n h2\n\n2mr\n∇2 + V(r)\n\n]\nψ(r) = ECMψ(r), (279)\n\nwhere ψ(r) is the wave function. The solution at large r (long time after\nthe collision) is sum of the incoming plane wave (ψin(r) = eik·r) and the\nscattered wave:\n\nψ(r) ∼\nr→∞ ψin(r) + f(χ)r eikr, (280)\n\nwhere k = mrvr/ h. Using this boundary condition, it can be shown that the\namplitude of the scattered waves can be written as a sum of partial waves:\n\nf(χ) =\n1\n\n2ik\n\n∞∑\nl=0\n\n(2l+ 1)(e2iηl − 1)Pl(cosχ), (281)\n\nwhere l is orbital quantum number, Pl is the Legendre polynomial of order\nl and ηl are phase shifts. The differential cross section is then determined\nfrom the amplitude of the scattered wave:\n\nIEL(χ) = |f(χ)|2 . (282)\n\nsemi-classical approximation To calculate the phase shifts ηl, of-\nten the so-called semi-classical approximation is used. In this approximation\nand in the short wave-length limit, the phase shifts are evaluated using WKB\n(Wentzel-Kramers-Brillouin) method [43]:\n\nηl =\n1\n\n4\nπ+\n\n1\n\n2\nlπ− kr0 +\n\n∞∫\nr0\n\n(κ(r) − k)dr, (283)\n\nwhere r0 is the turning point and κ(r) is the wave vector at the position r\n(k = κ(−∞)):\n\nκ(r) =\n(2mr)\n\n1\n2\n\n h\n(ECM − Veff(r))\n\n1\n2 , (284)\n\nwhere the effective potential Veff is given by:\n\nVeff = V(r) +\nl(l+ 1)\n\n2mrr2\n. (285)\n\n197\n\n\n\n198 elastic scattering in quantum mechanics\n\nIn this case (in the short wavelength limit), the semi-classical phase shift of\nEq. (283) can be related to the classical deflection function, via the so-called\nclassical equivalence relation (see Section 2.5):\n\nb =\nl+ 1\n\n2\n\nk\nand χ(b) = 2\n\n∂ηl\n∂l\n\n=\n2\n\nk\n\n∂ηl\n∂b\n\n. (286)\n\nThe last relation is valid only when l sinχ� 1 [41].\nWhen the classical equivalence relation is valid and when there is a sin-\n\ngle impact parameter contributing to the cross section at angle χ, the semi-\nclassical and classical differential cross sections will be the same. If there are\nmultiple impact parameters bi producing the same angle χ (as, for example,\nin the rainbow scattering of Section 2.4), the amplitudes of the scattered wave\nfi(χ) should first be summed and then squared to obtain the semi-classical\ndifferential cross section:\n\nIEL(χ) =\n\n∣∣∣∣∣∑\ni\n\nfi(χ)\n\n∣∣∣∣∣\n2\n\n. (287)\n\nSince we are summing the amplitudes and not the probabilities (squared\namplitudes), the cross section exhibits interference effects (oscillations) due\nto a phase difference between the trajectories associated with each impact\nparameter. For example, if there are two impact parameters b1 and b2 pro-\nducing the same deflections χ, the semi-classical angular cross section would\nbe given by [74]:\n\nIEL (χ) = IEL,1 (χ) + IEL,2 (χ) (288)\n\n+ (IEL,1 (χ))\n1/2 (IEL,2 (χ))\n\n1/2 cos(β2 −β1), (289)\n\nwhere IEL,1 (χ) and IEL,2 (χ) are the differential cross sections for each im-\npact parameter b1 and b2, evaluated according to Eq. (282), and β1 and β2\nare phase angles associated with each trajectory. Classically, the cross section\nwould consist only of the first two terms: IEL,CL (χ) = IEL,1 (χ) + IEL,2 (χ).\n\n\n\nD\nE L E C T R O D E D E S I G N F O R T H E M I X E D\nT R A N S V E R S E - L O N G I T U D I N A L C O M P R E S S I O N\n\nAs introduced in Chapter 6, the electric field needed for the mixed transverse-\nlongitudinal compression stage takes the form:\n\nE =\n\n\nEx\n\nEy\n\n−Ez\nz\n|z|\n\n , (290)\n\nwhere Ex, Ey and Ez are constants (with typically Ex = Ey, as in the trans-\nverse compression stage). The electric potential that produces the desired\nelectric field is given by:\n\nV(x,y, z) = V0 − Exx− Eyy+ Ez |z| , (291)\n\nwhere V0 is the potential at the origin O of the coordinate system (0, 0, 0),\nplaced at the target center (see Fig. 137). If we place the ground (V = 0) at\nthe tip of the target (at (x0/2, 0, 0)), the potential in the center of the target\nis:\n\nV0 = Ex\nx0\n2\n\n, (292)\n\nfor the target geometry as in Fig. 137.\n\nFigure 137: Target geometry, inside which we want to define the electric field given\nby Eq. (290). The origin of the coordinate system O coincides with the\ncenter of the target.\n\nStarting from this potential, we can obtain the shapes of the electrodes at\nthe target walls, that will produce the electric field of Eq. (290) inside the\n\n199\n\n\n\n200 electrode design for the mixed transverse-longitudinal compression\n\nFigure 138: Equipotential lines on the walls of the target of Fig. 137, given by\nEq. (291) and assuming Ex = Ey = 3Ez and the maximum potential\nof 1 kV. The coordinate l used to describe the equipotential lines on the\nvarious target wall is also indicated. The coordinate l = 0 corresponds\nto the bottom corner of the target.\n\ntarget. The shape and dimensions of the target are shown in Fig. 137. In the\nfollowing, we will obtain the expressions describing the equipotential lines\nat the three target walls separately (bottom, side and top walls). The desired\nelectric field is then created by placing electrodes with the same shape as\nthe obtained equipotential lines on the walls of the target and setting them\nto appropriate potentials, as shown in Fig. 138. The obtained electrodes for\nall three walls are first printed on a single Kapton foil (see Fig. 139), which\nis then folded into the triangular shape of the target.\n\nFigure 139: Equipotential lines of Fig. 138 flattened to a 2D surface. The coordinate\nsystem l− z, used to describe equipotential line shapes is also indicated.\nThe equipotential lines have different angles (α, β, γ) and spacing on\nthe different target walls (top, side, bottom).\n\n\n\nelectrode design for the mixed transverse-longitudinal compression 201\n\nbottom wall To obtain the equipotential lines on the bottom wall of\nthe target, we first need to find the expression for the electric potential in the\nplane of the bottom wall. The plane of the bottom target wall is described\nby the equation:\n\ny(x) = −\ny0\n4\n\n+ tan θ · x, (293)\n\nwhere y0 is the target height and 2θ is the target opening angle, as indicated\nin Fig. 137. Inserting this into Eq. (291) we get:\n\nV(x,y, z) = V0 + Ey\ny0\n4\n\n− (Ex + Ey tan θ) · x+ Ez |z| . (294)\n\nNotice that the dependence on y has disappeared. At this point it is helpful\nto introduce the coordinate l, which runs along the walls of the target as\nshown in Figs. 138 and 139. The x-coordinate can be related to l via the\nfollowing relation:\n\nx = l cos θ−\nx0\n2\n\n. (295)\n\nInserting this into Eq. (294) we get:\n\nV(l, z) = V0 + Ex\nx0\n2\n\n+ Ey(\ny0\n4\n\n+\nx0\n2\n\ntan θ)\n\n− (Ex cos θ+ Ey sin θ) · l+ Ez |z| (296)\n\nNotice that the term in from of l in Eq. (296) is simply a sum of projections\nof the x-component of the electric field and the y-component of the electric\nfield onto the bottom wall of the target (Ex cos θ and Ey sin θ, respectively).\nWe can simplify Eq. (296) by grouping all the constant terms in front into a\npotential V1:\n\nV1 ≡ V0 + Ex\ny0\n4\n\n+ Ex\nx0\n2\n\n+ Ey tan θ\nx0\n2\n\n. (297)\n\nBy noting that V0 = Ex x02 and tan θ = y0\n2x0\n\n, we get that:\n\nV1 = Exx0 + Ey\ny0\n2\n\n, (298)\n\nwhich is exactly the potential in the bottom corner of the target (at l = z = 0).\nThus, Eq. (296) becomes:\n\nV(l, z) = V1 − (Ex cos θ+ Ey sin θ) · l+ Ez |z| . (299)\n\nAn equipotential line is obtained by requiring the potential to be equal to a\nfixed value Vc:\n\nV(l, z) = Vc, (300)\n\nwhich combined with the Eq. (299) gives us:\n\nVc = V1 − Ex(cos θ+ sin θ) · l+ Ez |z| . (301)\n\n\n\n202 electrode design for the mixed transverse-longitudinal compression\n\nThus, the equation describing the equipotential lines on the bottom wall of\nthe target is a simple linear function:\n\nz(l) =\n\n{\nV1−Vc\nEz\n\n− (ExEz cos θ+ Ey\nEz\n\nsin θ) · l for z < 0\n\n−V1−VcEz\n+ (ExEz cos θ+ Ey\n\nEz\nsin θ) · l for z > 0,\n\n(302)\n\nwhich can be rewritten in terms of the equipotential line angle γ, defined in\nFig. 139:\n\nz(l) =\n\n{\nV1−Vc\nEz\n\n− tanγ · l for z < 0\n\n−V1−VcEz\n+ tanγ · l for z > 0,\n\n(303)\n\nwhere tanγ is given by the following relation:\n\ntanγ =\nEx\n\nEz\ncos θ+\n\nEy\n\nEz\nsin θ. (304)\n\nSeveral equipotential lines, spaced by 25 V, are shown in Figs. 138 and 139,\nassuming Ex = Ey = 3Ez.\n\nFrom Eq. (303) it follows that the distance in the z-direction between two\nequipotential lines with potential difference ∆V is:\n\n∆z =\n∆V\n\nEz\n, (305)\n\nwhile the distance in the l-direction is:\n\n∆l =\n∆V\n\nEz\n\n1\n\ntanγ\n=\n\n∆z\n\ntanγ\n. (306)\n\nThe distance perpendicular to the equipotential line (the minimum distance)\nis then:\n\nd = ∆z cosγ =\n∆V\n\nEz\ncosγ. (307)\n\nFrom these equations we can thus determine the electrode spacing on the\nbottom wall of the target.\n\nside wall The plane of the target side wall is described by the equation:\n\nx = −\nx0\n2\n\n. (308)\n\nInsert this into Eq. (291) we get:\n\nV(x,y, z) = V0 + Ex\nx0\n2\n\n− Eyy+ Ez |z| (309)\n\nWe can rewrite this in terms of the coordinate l by noting that for the side\nwall y = −l− y0\n\n2 (see Figs. 138 and 139):\n\nV(l, z) = V0 + Ex\nx0\n2\n\n+ Ey\ny0\n2\n\n+ Ey · l+ Ez |z| (310)\n\n= V1 + Ey · l+ Ez |z| , (311)\n\n\n\nelectrode design for the mixed transverse-longitudinal compression 203\n\nwhere we have used the definition of potential V1 of Eq. (298). The equipo-\ntential line (at a fixed potential Vc) is then given by:\n\nz(l) =\n\n{\nV1−Vc\nEz\n\n+\nEy\nEz\n· l for z < 0\n\n−V1−VcEz\n−\n\nEy\nEz\n· l for z > 0,\n\n(312)\n\nor written in terms of the equipotential line angle β:\n\nz(l) =\n\n{\nV1−Vc\nEz\n\n+ tanβ · l for z < 0\n\n−V1−VcEz\n− tanβ · l for z > 0.\n\n(313)\n\nThus we can identify that the equipotential line angle is given by:\n\ntanβ =\nEy\n\nEz\n. (314)\n\nSimilar as before, from Eq. (313) we can find the distances between two\nequipotential lines with voltage drop ∆V between them:\n\n∆z =\n∆V\n\nEz\n(315)\n\n∆l =\n∆z\n\ntanβ\n(316)\n\nd = ∆z cosβ, (317)\n\nwhere ∆z and ∆l are the distances in the z and l-directions, respectively,\nand d is the distance perpendicular to the equipotential lines (the minimum\ndistance).\n\ntop wall The plane of the top target wall is described by the equation:\n\ny(x) =\ny0\n4\n\n− tan θ · x, (318)\n\nwhich leads to the following equation for the potential at the top wall:\n\nV(x,y, z) = V0 − Ey\ny0\n4\n\n− (Ex − Ey tan θ) · x+ Ez |z| (319)\n\nor written in terms of the coordinate l, which is in this case related to the\nx-coordinate via x = −(l+ y0) cos θ− x0\n\n2 (see Figs. 138 and 139):\n\nV(l, z) = V0 + Ex\nx0\n2\n\n− Ey(\ny0\n4\n\n+\nx0\n2\n\ntan θ)\n\n− (Ex cos θ− Ey sin θ) · (l+ y0) + Ez |z| (320)\n\n= V2 − (Ex cos θ− Ey sin θ) · (l+ y0) + Ez |z| , (321)\n\nwhere we defined a potential V2 = V1 − Eyy0, which is exactly the potential\nat the top corner of the target (at z = 0, l = −y0). Thus, the equipotential\nline of a fixed potential Vc is then given by:\n\nz(l) =\n\n{\nV2−Vc\nEz\n\n+ (ExEz cos θ− Ey\nEz\n\nsin θ) · (l+ y0) for z < 0\n\n−V2−VcEz\n− (ExEz cos θ− Ey\n\nEz\nsin θ) · (l+ y0) for z > 0,\n\n(322)\n\n\n\n204 electrode design for the mixed transverse-longitudinal compression\n\nor in terms of equipotential line angle α:\n\nz(l) =\n\n{\nV2−Vc\nEz\n\n+ tanα · (l+ y0) for z < 0\n\n−V2−VcEz\n− tanα · (l+ y0) for z > 0,\n\n(323)\n\nwith the equipotential line angle:\n\ntanα =\nEx\n\nEz\ncos θ−\n\nEy\n\nEz\nsin θ. (324)\n\nFrom Eq. (323) we can obtain the distances between two equipotential lines\nwith voltage drop ∆V between them:\n\n∆z =\n∆V\n\nEz\n(325)\n\n∆l =\n∆z\n\ntanα\n(326)\n\nd = ∆z cosα. (327)\n\n\n\nA C K N O W L E D G M E N T S\n\nFirst, I would like to thank Klaus Kirch for offering me the opportunity\nto pursue my PhD in his group, and for his continuous support, always\nconstructive suggestions and many opportunities to participate in various\nschools and conferences. I am especially grateful to my day-to-day supervi-\nsor Aldo Antognini for numerous discussions, and for all the effort and the\nsupport throughout my whole PhD and the thesis writing process.\n\nI have greatly enjoyed working with Andreas Eggenberger - especially all\nthe discussions, not just about work, but also many other topics. I think this\nhas created very nice working atmosphere during the beam-times and the\nrest of the time (especially with all the mimics and degeneration with Aldo\nand you).\n\nI am thankful for always precise mechanical parts of Bruno Zehr from\nthe IPA workshop, even when ordered at the very last minute. I am also\ngrateful to Urs Greuter and Claudio Kämpf for allowing me to use the reflow\nsoldering equipment and for the help provided with it. The success of all\nthe measurements would never have been possible without the excellent\nsupport of the PSI detector group, especially Malte Hildebrandt and Alexey\nStoykov.\n\nI would also like to thank the former and current members of the muCool\ncollaboration: Angela Papa (for all the fun night shifts and late dinners - easy\nfor us!), Nick Ayres (for proofreading my thesis and fixing all those pesky\narticles), Vira Bondar (for all the chocolate and nice words), David Taqqu\n(for his brilliant ideas and the recent discussion about Heµ+ ion formation),\nKim Siang Khaw (for the first introduction to Geant4 and serial device read-\nout with Python), Ryoto Iwai (for investing so much effort to test my crazy\nidea), Andreas Knecht (for all the help and patience during the looong wait-\ning times), Jonas Nuber, Hans-Christian Koch, Claude Petitjean and all the\nothers that contributed to the experiment.\n\nI am also grateful to the former and current members of our group at ETH\nand the muon group at PSI: Michał Rawlik (for all the help you provided at\nthe probably most difficult point of my PhD), mademoiselle Laura Sinku-\nnaite (for listening to all my complaints, for all the skiing and hiking trips\nand most importantly, for the famous via ferrata without which I would\nhave never finished writing), Patrick Schwendimann (for translating my ab-\nstract to German, and for all the delicious cakes and ice-cream - meow!), Mr.\nMiroslaw Marszalek (for all the spontaneous board game evenings), Karsten\nSchumann (for all the relevant and not so relevant, but always interesting\nfacts), Stella Vogiatzi, Alex Skawran, Narongrit Ritjoho, Solange Emmeneg-\nger, Manuel Zeyen and all the others that created the friendly atmosphere\nand supported me, especially during this last difficult part of the writing.\n\nI am grateful to Matija, without whom I might have never ended up at\nETH at all. 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(Butterworths, London,\n1972).\n\nhttp://dx.doi.org/10.1088/0022-3700/12/24/024\nhttp://dx.doi.org/10.1088/0022-3700/12/24/024\nhttp://dx.doi.org/https://doi.org/10.1016/B978-008044479-6.50004-2\nhttp://dx.doi.org/10.1007/0-306-47112-4_84\nhttp://dx.doi.org/10.1049/ip-a-1.1980.003\nhttp://dx.doi.org/10.1049/ip-a-1.1980.003\n\n\tAbstract\n\tZusammenfassung\n\tContents\n\t1 Introduction\n\t1.1 Working principle\n\t1.2 Status of the muCool project\n\t1.3 Outline of the thesis\n\n\t2 Theory of muon-helium collisions\n\t2.1 Introduction\n\t2.2 Classical scattering theory\n\t2.3 High-energy regime\n\t2.4 Low-energy regime\n\t2.5 Summary and validity of the classical scattering theory\n\n\t3 Geant4 simulations of the muon stopping and drift in helium gas\n\t3.1 Implementation of low-energy muon-He interactions in Geant4\n\t3.2 Muon range and slowing-down in He gas\n\t3.3 Muon drift in an uniform electric field\n\t3.4 Muon drift in crossed electric and magnetic fields\n\t3.5 Conclusions\n\n\t4 Transverse compression\n\t4.1 Setup for the transverse compression\n\t4.2 Results of the 2015 beam-time\n\t4.3 Electric field scan \n\t4.4 Conclusions\n\n\t5 Longitudinal compression\n\t5.1 Improved test of the longitudinal compression \n\t5.2 Additional muon losses?\n\t5.3 Longitudinal compression with E B drift\n\t5.4 Conclusions\n\n\t6 Mixed transverse-longitudinal compression\n\t6.1 Working principle\n\t6.2 Electric field design\n\t6.3 Measurements and results of the 2017 beam-time\n\t6.4 Conclusions\n\n\t7 Next steps\n\t7.1 Introduction\n\t7.2 Target with cold extension\n\t7.3 Target with warm extension\n\t7.4 Conclusions\n\n\t8 Summary and outlook\n\tAppendix\n\tA Motion of charged particles in crossed electric and magnetic fields in the vacuum\n\tB Elastic collisions\n\tC Elastic scattering in quantum mechanics\n\tD Electrode design for the mixed transverse-longitudinal compression\n\tAcknowledgements\n\n\tAcknowledgments\n\tBibliography"},"fulltext":true,"key":"53e8a9b8a222fd9a0d1ac5d67a5904d5","url":"https://inspirehep.net/files/53e8a9b8a222fd9a0d1ac5d67a5904d5"}],"citation_count_without_self_citations":2,"authors":[{"raw_affiliations":[{"value":"Zurich, ETH"}],"affiliations_identifiers":[{"schema":"ROR","value":"https://ror.org/05a28rw58"}],"full_name_unicode_normalized":"belosevic, ivana","full_name":"Belosevic, Ivana","record":{"$ref":"https://inspirehep.net/api/authors/2552945"},"affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/903369"},"value":"Zurich, ETH"}],"ids":[{"schema":"INSPIRE BAI","value":"I.Belosevic.2"}],"last_name":"Belosevic","signature_block":"BALASAFACi","first_name":"Ivana","uuid":"57e9e0fd-03fe-4797-996f-029cd06e22c0","recid":2552945}],"citation_count":3,"$schema":"https://inspirehep.net/schemas/records/hep.json","keywords":[{"source":"author","value":"MUONS (PARTICLE PHYSICS)"},{"source":"author","value":"Muon beam"},{"source":"author","value":"Phase space compression"},{"source":"author","value":"Phase space cooling"},{"source":"author","value":"Geant4 simulations"}],"references":[{"reference":{"publication_info":{"journal_volume":"84","artid":"73","year":2015,"page_start":"73","journal_title":"Prog.Part.Nucl.Phys."},"title":{"title":"Precision muon physics"},"misc":["1T. 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The averaged F1 score of almost 90 percent implies the\n possibility of categorising the type of radiators from their magnitude only\n emitted near-field pattern"}],"control_number":1859233,"earliest_date":"2021-03-16","document_type":["article"],"texkeys":["Geranmayeh:2021hla"],"abstracts":[{"source":"arXiv","value":"Sets of intelligent classifiers are applied to the near-field scan-data in order to automatically classify the shape of radiating wirings. The support vector machine, k-nearest neighbors algorithm, and Gaussian process classifications are trained using the near-field radiation pattern of diverse radiating wire configurations. Leave-one-out cross-validation is used for estimating the performance of the predictive models. The output of this research is a software package well-suited to be retrained based on any measured near-field databank to automate the identification of magnetic-type or electric-type of the radiating coupling sources.","abstract_source_suggest":{"input":"arXiv"}}],"primary_arxiv_category":["eess.IV"],"titles":[{"source":"arXiv","title":"Machine-Learning Classification of Closed and Open Radiating Wires from Near Magnetic or Electric Field Scan Images"}],"facet_author_name":["2518738_Amir Geranmayeh"],"license":[{"license":"CC BY 4.0","material":"preprint","url":"http://creativecommons.org/licenses/by/4.0/"}],"_oai":{"sets":["Literature"],"id":"oai:inspirehep.net:1859233","updated":"2023-03-06T15:03:56.301464"},"curated":false,"arxiv_eprints":[{"categories":["eess.IV","cs.LG","physics.app-ph","physics.ins-det"],"value":"2104.09277"}]},"updated":"2023-03-06T15:03:56.301464+00:00","created":"2021-04-20T00:00:00+00:00","id":"1859233","links":{"bibtex":"https://inspirehep.net/api/literature/1859233?format=bibtex","latex-eu":"https://inspirehep.net/api/literature/1859233?format=latex-eu","latex-us":"https://inspirehep.net/api/literature/1859233?format=latex-us","json":"https://inspirehep.net/api/literature/1859233?format=json","json-expanded":"https://inspirehep.net/api/literature/1859233?format=json-expanded","cv":"https://inspirehep.net/api/literature/1859233?format=cv","citations":"https://inspirehep.net/api/literature/?q=refersto%3Arecid%3A1859233"}},{"metadata":{"documents":[{"filename":"document.pdf","attachment":{"content":"HAL Id: tel-02494060\nhttps://tel.archives-ouvertes.fr/tel-02494060v2\n\nSubmitted on 28 Feb 2020\n\nHAL is a multi-disciplinary open access\narchive for the deposit and dissemination of sci-\nentific research documents, whether they are pub-\nlished or not. The documents may come from\nteaching and research institutions in France or\nabroad, or from public or private research centers.\n\nL’archive ouverte pluridisciplinaire HAL, est\ndestinée au dépôt et à la diffusion de documents\nscientifiques de niveau recherche, publiés ou non,\némanant des établissements d’enseignement et de\nrecherche français ou étrangers, des laboratoires\npublics ou privés.\n\nOptimization of future projects for the measurement of\nCosmic Microwave Background polarization\n\nThuong Hoang Duc\n\nTo cite this version:\nThuong Hoang Duc. Optimization of future projects for the measurement of Cosmic Microwave\nBackground polarization. Physics [physics]. Université Sorbonne Paris Cité, 2018. English. �NNT :\n2018USPCC235�. �tel-02494060v2�\n\nhttps://tel.archives-ouvertes.fr/tel-02494060v2\nhttps://hal.archives-ouvertes.fr\n\n\nUNIVERSITÉ SORBONNE PARIS CITÉ\n\nUNIVERSITÉ PARIS DIDEROT\n\nÉcole doctorale des Sciences de la Terre et de l’Environnement\n\net Physique de l’Univers (STEP’UP), Paris - ED 560\n\nLaboratoire Astroparticule et Cosmologie (APC)\n\nOptimization of future projects for the measurement of\nCosmic Microwave Background polarization\n\nHOANG Duc Thuong\n\nThèse de doctorat d’Université Physique de l’Univers\n\ndirigée par Guillaume Patanchon\n\nprésentée et soutenue publiquement le 17 décembre 2018\n\ndevant un jury composé de:\n\nGérard Rousset Université Paris Diderot, LESIA Président du jury\nFrançois Couchot CNRS, LAL Rapporteur\nGiampaolo Pisano Cardiff University Rapporteur\nSophie Henrot-Versillé CNRS, LAL Examinatrice\nGuillaume Patanchon Université Paris Diderot, APC Directeur de thèse\nDamien Prêle CNRS, APC Co-encadrant\n\nhttp://www.sorbonne-paris-cite.fr/\nhttp://www.apc.univ-paris7.fr/APC_CS/\nhttp://www.univ-paris-diderot.fr\nhttp://ed560.ipgp.fr/index.php/Accueil\nhttp://www.sorbonne-paris-cite.fr/\nhttps://www.usth.edu.vn/en/\nhttp://ed560.ipgp.fr/index.php/Accueil\nhttp://ed560.ipgp.fr/index.php/Accueil\nhttp://ed560.ipgp.fr/index.php/Accueil\nhttp://ed560.ipgp.fr/index.php/Accueil\n\n\nAbstract i\n\nAbstract\n\nDuring my Ph.D., my research focused on the development of future projects for\nthe measurement of Cosmic Microwave Background (CMB) polarization aimed\nto probe primordial B mode. Achieving this goal will not only require sufficient\ndetector array sensitivity but also unprecedented control of all systematic errors\ninherent to CMB polarization measurements. One of the important effects is\nthe bandpass mismatch error which is the effect of non-uniformity or mismatch\nof the bandpass filters for different detectors inducing leakage from foreground\nintensity to polarization after calibrating the data on CMB. I estimated the level\nof the leakage for a realistic configuration of the forthcoming LiteBIRD JAXA\nmission with simulation and found that the amplitude of leakage depends on the\nscanning strategy of the satellite parameterized with precession angle, spin angle,\nprecession and rotation velocities. After the study, I proposed some nearly optimal\nconfigurations to archive the target of tensor-to-scalar ratio r. The bias from\nforeground leakage in the range 2 ≤ ` ≤ 10 (reionization bump) is of the order of\nabout 5× 10−4 and in the range 10 ≤ ` ≤ 200 (recombination bump) of the order\nof about 5× 10−5.\n\nThe second topic of my thesis was an instrumental study: the interaction of par-\nticles with a Transition Edge Sensors (TES) array using the focal plane of the\nground-based QUBIC (Q & U Bolometric Interferometer for Cosmology) experi-\nment. The goal of this work was to test the behaviour of detectors to cosmic rays\n(such as time-constants and cross-talk). I placed an Americium 241 radioactive\nsource in front of a 256 TESs array inside a cryostat. When particles hit one\nof the components of a pixel (eg: Thermometer, absorbing grid, substrate), the\ndeposited energy induced temperature elevation among components and possibly\nto the neighbor pixels. This could provide an evaluation of the cross-talk between\npixels. Moreover, this study allows us to understand the thermal and electronic\nreadout system time constants of a TES.\n\nKeywords: Cosmic Microwave Background polarization, CMB experiments, Data\nanalysis, instrumentation, Observational cosmology\n\n\n\nRésumé ii\n\nAu cours de ma thèse, mes recherches ont porté sur le développement des projets\nfuturs de mesure de la polarisation du fond diffus cosmologique (CMB) visant\nà sonder les modes B primordiaux. Pour atteindre cet objectif, il faudra non\nseulement une sensibilité suffisante des matrices de détecteurs, mais également un\ncontrôle sans précédent de toutes les erreurs systématiques inhérentes aux mesures\nde polarisation du CMB. Une source importante d’erreur systématique est la non-\nuniformité ou la non-concordance des filtres passe-bande des différents détecteurs.\nCet effet induit des fuites de l’intensité vers la polarisation après l’étalonnage\ndes données. J’ai estimé le niveau de fuite pour une configuration réaliste de\nla prochaine mission LiteBIRD de la JAXA à l’aide de simulations et montré que\nl’amplitude de la fuite dépendait de la stratégie de balayage du satellite paramétrée\npar l’angle de rotation du satellite, l’angle de précession et les vitesses de précession\net de rotation. En conclusion de cette étude, j’ai proposé des configurations quasi\noptimales pour LiteBIRD permettant d’atteindre l’objectif sur le rapport tenseur\nsur scalaire r. Le biais dû à la fuite des avant-plans dans l’intervalle 2 ≤ ` ≤ 10\n\n(bosse de réionisation) est de l’ordre de 5× 10−4 et dans l’intervalle 10 ≤ ` ≤ 200\n\n(bosse de recombinaison) de l’ordre de 5× 10−5.\n\nLe deuxième sujet de ma thèse était une étude instrumentale : l’interaction des\nparticules avec une matrice de TES. Pour ce faire, j’ai utilisé le plan focal de\nl’expérience QUBIC (interféromètre bolométrique Q & U pour la cosmologie). Le\nbut de ce travail était de tester le comportement des détecteurs aux rayons cos-\nmiques (tels que les constantes de temps et la diaphonie entre détecteurs). J’ai\nplacé une source radioactive d’américium 241 devant un réseau de 256 TES à\nl’intérieur d’un cryostat. Lorsque les particules interagissent avec l’un des com-\nposants d’un pixel (ex: thermomètre, grille absorbante, substrat), l’énergie déposée\nprovoque une élévation de la température d’un composant et éventuellement celui\nd’un pixel voisin. Cela pourrait fournir une évaluation de la diaphonie entre pix-\nels. De plus, cette étude nous permet de comprendre les constantes de temps du\nsystème de lecture thermique et électronique d’un TES.\n\nMots clés: Polarisation du fond diffus cosmologique, expériences CMB, analyse\nde données, instrumentation, cosmologie observationnelle\n\n\n\nAcknowledgements\n\nThe last three years at the AstroParticle and Cosmology (APC) laboratory have\nbeen a time of massive learning and growth for my experiences, both personally\nand professionally. This thesis would not have been possible without the guid-\nance, support, international collaborations, and motivation from my supervisors,\ncolleagues, friends, family and my willpower.\n\nFirstly, I would like to express from my heart my deepest gratitude to my su-\npervisor Dr. Guillaume Patanchon for the continuous support of my Ph.D study\nand related research, for his patience, motivation, and immense knowledge. His\nguidance and discussion helped me in all the time of research and writing of this\nthesis. As well as, I would like to express my sincere thanks to my co-supervisor\nDr. Damien Prêle for his scientific guidance and encouraged supports to widen\nmy research from various perspectives.\n\nI am also indebted to Dr. Giraud-Héraud Yannick for many administrative proce-\ndures, many helpful comments on the chapter: interaction of particles with a TES\narray, he is an unofficial advisor during my thesis.\n\nMy sincere thanks also goes to Prof. Jean Christophe Hamilton, Prof. Michel\nPiat, who provided me an opportunity to join their team, and who gave access to\nthe laboratory and research facilities. Without they precious support it would not\nbe possible to conduct this research. Many difficult problems in instrumentation\nare solved after discussions with Michel Piat, he is like my technical advisor since\nI have been studying TES.\n\nBesides my supervisors, I would like to thank the rest of my thesis committee:\nProf. Gérard Rousset, Dr. Giampaolo Pisano, and Dr. François Couchot, Dr.\nSophie Henrot-Versillé for their insightful comments and encouragement.\n\nI would like to thank my working team on bandpass mismatch systematic effect\nstudy Martin Bucher, Tomotake Matsumura, Ranajoy Banerji, Hirokazu Ishino,\nMasashi Hazumi, Jacques Delabrouille. I gain my knowledge with many scientific\ndiscussions in the team. With them I have the first publication of my scientific\ncareer, it is always memorable.\n\niii\n\n\n\niv\n\nI would like to thank to Ken Ganga, Cyrille Rosset, Fabrice Voisin, Laurent Grand-\nsire, Steve Torchinsky, Matthieu Tristram, Joseph Martino and all members of\nAPC laboratory for many scientific discussions.\n\nI thank my fellow labmates (Alessandro, Bastien, Cyrille, Camille, Clara, Dominic,\nEleonora, Calum, Jie, Mikhail, Maria, Louise, Pierros, Si, Thomas, Tuan) for the\nstimulating discussions, for the sleepless nights we were working together before\ndeadlines, and for all the fun we have had in the last three years. I am very\nthankful to my international friends as well as Vietnamese friends throughout my\nunforgettable stay in Paris.\n\nLast but not the least, I would like to thank my family: my parents and to my\nbrother for supporting me spiritually throughout doing this thesis and my life in\ngeneral.\n\n\n\nv\n\nDedicated to my family\n\n\n\nContents\n\nAbstract i\n\nRésumé ii\n\nAcknowledgements iii\n\nContents vi\n\nList of Figures ix\n\nList of Tables xiii\n\nAbbreviations xiv\n\n1 Introduction 1\n1.1 Evaluating the level of the bandpass mismatch systematic effect for\n\nthe future CMB satellites . . . . . . . . . . . . . . . . . . . . . . . . 1\n1.2 Interaction of particles with a TES array . . . . . . . . . . . . . . . 2\n\n2 Introduction to cosmology 5\n2.1 The Hot Big Bang theory . . . . . . . . . . . . . . . . . . . . . . . 6\n2.2 The standard cosmological model . . . . . . . . . . . . . . . . . . . 15\n2.3 Expanding Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . 16\n2.4 General relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n\n2.4.1 Friedmann–Lemaître–Robertson–Walker (FLRW) metric . . 18\n2.4.2 Geodesic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n2.4.3 Einstein equations and Friedmann equations . . . . . . . . . 20\n\n2.5 Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27\n2.6 The horizon problem . . . . . . . . . . . . . . . . . . . . . . . . . . 28\n2.7 The flatness problem . . . . . . . . . . . . . . . . . . . . . . . . . . 30\n2.8 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30\n2.9 Physics of inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32\n\n2.9.1 Slow-Roll inflation . . . . . . . . . . . . . . . . . . . . . . . 34\n2.10 Primordial quantum fluctuations in inflation and cosmological per-\n\nturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36\n2.10.1 Linear perturbation . . . . . . . . . . . . . . . . . . . . . . . 37\n\nvi\n\n\n\nContents vii\n\n2.10.2 Primordial quantum fluctuations in inflation . . . . . . . . . 38\n2.10.3 Cosmological perturbations and structure formation . . . . . 43\n\n3 The Cosmic Microwave Background (CMB) 46\n3.1 The CMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49\n3.2 Physics of CMB temperature anisotropies . . . . . . . . . . . . . . . 60\n3.3 CMB polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63\n3.4 Primordial non-Gaussianity in the CMB . . . . . . . . . . . . . . . 68\n3.5 Gravitational lensing . . . . . . . . . . . . . . . . . . . . . . . . . . 69\n3.6 CMB spectral distortions . . . . . . . . . . . . . . . . . . . . . . . . 70\n3.7 Foreground components . . . . . . . . . . . . . . . . . . . . . . . . 72\n\n3.7.1 Thermal dust . . . . . . . . . . . . . . . . . . . . . . . . . . 74\n3.7.2 Synchrotron . . . . . . . . . . . . . . . . . . . . . . . . . . . 75\n3.7.3 Free-free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76\n3.7.4 Spinning dust . . . . . . . . . . . . . . . . . . . . . . . . . . 76\n\n3.8 Systematic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76\n3.8.1 Cosmic rays . . . . . . . . . . . . . . . . . . . . . . . . . . . 78\n3.8.2 Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78\n3.8.3 Bandpass mismatch . . . . . . . . . . . . . . . . . . . . . . . 80\n\n3.9 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80\n3.10 QUBIC and LiteBIRD . . . . . . . . . . . . . . . . . . . . . . . . . 83\n\n3.10.1 Ground base experiment: QUBIC . . . . . . . . . . . . . . . 83\n3.10.1.1 General principle . . . . . . . . . . . . . . . . . . . 83\n3.10.1.2 Instrument . . . . . . . . . . . . . . . . . . . . . . 87\n\n3.10.2 Space satellite mission: LiteBIRD . . . . . . . . . . . . . . . 88\n\n4 Band-pass mismatch 96\n4.1 Sky emission model and mismatch errors . . . . . . . . . . . . . . . 98\n4.2 Calculating the bandpass mismatch . . . . . . . . . . . . . . . . . . 104\n\n4.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107\n4.2.2 Analytic estimates . . . . . . . . . . . . . . . . . . . . . . . 119\n4.2.3 Precession period and spin period ratio τprec/τspin . . . . . . 125\n\n4.3 A correction method . . . . . . . . . . . . . . . . . . . . . . . . . . 133\n4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136\n\n5 Interaction of particles with a TES array 137\n5.1 Theory of a superconducting Transition-Edge Sensor . . . . . . . . 139\n\n5.1.1 Theory of superconductivity . . . . . . . . . . . . . . . . . . 139\n5.1.2 The superconducting transition region . . . . . . . . . . . . 141\n5.1.3 Principle of a Transition-Edge Sensor (TES) . . . . . . . . . 142\n\n5.1.3.1 Electrical and thermal response . . . . . . . . . . . 142\n5.1.3.2 Noise performance . . . . . . . . . . . . . . . . . . 151\n\n5.2 TES arrays of the QUBIC experiment . . . . . . . . . . . . . . . . . 155\n5.3 The cryostat and the electronic readout system . . . . . . . . . . . 160\n\n5.3.1 IV, PV, RV curves . . . . . . . . . . . . . . . . . . . . . . . 173\n\n\n\nContents viii\n\n5.4 Radioactive source Americium 241 . . . . . . . . . . . . . . . . . . 176\n5.5 TES model approach . . . . . . . . . . . . . . . . . . . . . . . . . . 177\n5.6 Glitches data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 182\n\n5.6.1 Glitches detection . . . . . . . . . . . . . . . . . . . . . . . . 184\n5.6.2 Fit glitches . . . . . . . . . . . . . . . . . . . . . . . . . . . 190\n5.6.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . 195\n5.6.4 Time constants and the KI parameter of the PID controller 198\n5.6.5 Time constants, amplitude and the voltage bias VTES . . . . 201\n\n5.7 Cross-talk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203\n5.7.1 Thermal cross-talk . . . . . . . . . . . . . . . . . . . . . . . 203\n5.7.2 Cross-talk of the electronic readout chain . . . . . . . . . . . 209\n\n5.8 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . . . 211\n\n6 Conclusion and perspectives 213\n\nA Solutions of Einstein equations 216\nA.1 FLRW solution and Friedmann equations . . . . . . . . . . . . . . . 216\n\nA.1.1 Christoffel symbols for the FLRW metric . . . . . . . . . . . 217\nA.1.2 Ricci tensor and Einstein’s tensor . . . . . . . . . . . . . . . 217\n\nA.2 Stress-energy tensor Tµν . . . . . . . . . . . . . . . . . . . . . . . . 218\nA.3 Schwarzschild Solution and Black Holes . . . . . . . . . . . . . . . . 219\n\nB χ2 and fit C` 220\n\nC Fitted glitches 222\n\nBibliography 228\n\nPublications 245\n\n\n\nList of Figures\n\n2.1 D’où venons-nous? Que sommes-nous? Où allons-nous? . . . . . . . 6\n2.2 The evolution of the Universe . . . . . . . . . . . . . . . . . . . . . 7\n2.3 The early Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n2.4 The primordial abundance of light elements in the Universe . . . . . 11\n2.5 The accelerating Universe . . . . . . . . . . . . . . . . . . . . . . . 12\n2.6 The 2 dF galaxy redshift survey . . . . . . . . . . . . . . . . . . . . 13\n2.7 The first detection of gravitational waves . . . . . . . . . . . . . . . 14\n2.8 The standard cosmological model . . . . . . . . . . . . . . . . . . . 15\n2.9 The Hubble constant . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n2.10 The stress-energy tensor . . . . . . . . . . . . . . . . . . . . . . . . 21\n2.11 The fate of the Universe . . . . . . . . . . . . . . . . . . . . . . . . 24\n2.12 The evolution of scalar and time . . . . . . . . . . . . . . . . . . . . 25\n2.13 The horizon problem . . . . . . . . . . . . . . . . . . . . . . . . . . 29\n2.14 The inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33\n2.15 The inflation scenarios . . . . . . . . . . . . . . . . . . . . . . . . . 36\n2.16 The CMB anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . 37\n2.17 Evolution of photons, baryons and dark matter . . . . . . . . . . . 44\n\n3.1 The CMB temperature maps . . . . . . . . . . . . . . . . . . . . . . 51\n3.2 The CMB temperature angular power spectrum CTT\n\n` . . . . . . . . 56\n3.3 The last scattering surface . . . . . . . . . . . . . . . . . . . . . . . 57\n3.4 The physical mechanism of CMB temperature anisotropies . . . . . 61\n3.5 Temperature angular power spectrum and 4 cosmological parameters 62\n3.6 The Thomson scattering . . . . . . . . . . . . . . . . . . . . . . . . 63\n3.7 Quadrupolar anisotropies . . . . . . . . . . . . . . . . . . . . . . . . 64\n3.8 The pure E-mode and pure B-mode . . . . . . . . . . . . . . . . . . 66\n3.9 The temperature and polarization angular power spectra . . . . . . 67\n3.10 CMB lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70\n3.11 CMB spectral distortion . . . . . . . . . . . . . . . . . . . . . . . . 71\n3.12 foreground separation . . . . . . . . . . . . . . . . . . . . . . . . . . 72\n3.13 The micrawave sky components . . . . . . . . . . . . . . . . . . . . 73\n3.14 The Planck 2018 result of thermal dust polarization at 353 GHz . . 74\n3.15 The Planck 2018 result of syncrochon polarization at 30 GHz . . . . 75\n3.16 The Planck HFI systematic effect . . . . . . . . . . . . . . . . . . . 77\n3.17 The Planck data of cosmic rays . . . . . . . . . . . . . . . . . . . . 79\n\nix\n\n\n\nList of Figures x\n\n3.18 Spectral filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81\n3.19 The state of the art after Planck 2018 result . . . . . . . . . . . . . 82\n3.20 QUBIC self-calibration results . . . . . . . . . . . . . . . . . . . . . 84\n3.21 QUBIC cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86\n3.22 QUBIC instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . 87\n3.23 The concept design of the LiteBIRD spacecraft . . . . . . . . . . . 89\n3.24 The concept design of LiteBIRD and the layout of focal plane unit . 89\n3.25 LiteBIRD frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . 92\n3.26 The low/mid frequency technology . . . . . . . . . . . . . . . . . . 93\n3.27 The high-frequency technology . . . . . . . . . . . . . . . . . . . . . 94\n\n4.1 The satellite scanning strategy . . . . . . . . . . . . . . . . . . . . . 97\n4.2 Simulation pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . . 99\n4.3 Input intensity I map containing CMB and thermal dust at 140GHz 100\n4.4 The simulation of bandpass function . . . . . . . . . . . . . . . . . 102\n4.5 The mid-frequency focal plane . . . . . . . . . . . . . . . . . . . . . 105\n4.6 Hitcount map for a fiducial scanning strategy . . . . . . . . . . . . 108\n4.7 Q and U leakage maps . . . . . . . . . . . . . . . . . . . . . . . . . 109\n4.8 Mask 20 % sky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110\n4.9 EE and BB leakage power spectra for α = 65◦, β = 30◦, τspin =\n\n10min, τprec= 96.1803min . . . . . . . . . . . . . . . . . . . . . . . 111\n4.10 BB leakage power spectra . . . . . . . . . . . . . . . . . . . . . . . 112\n4.11 BB leakage power spectra for different scanning parameters. . . . . 113\n4.12 The angular power spectrum of 10 realizations . . . . . . . . . . . . 114\n4.13 The angular power spectrum of 2 % variation of the filters . . . . . 115\n4.14 The global offset across the focal plane . . . . . . . . . . . . . . . . 116\n4.15 Angular power spectrum of the global offset across the focal plane . 117\n4.16 The leakage map of pair detector with nominal locations . . . . . . 117\n4.17 The BB angular power spectrum for each pair detector . . . . . . . 118\n4.18 The hitcount map and leakage maps of the Planck scanning strategy\n\ncase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119\n4.19 Power spectrum of the Planck scanning strategy case . . . . . . . . 120\n4.20 EE and BB leakage power spectra with rotating HWP . . . . . . . 121\n4.21 Leakage for the Q component relative to the dust temperature\n\n(δQ/IGal) using one bolometer pair . . . . . . . . . . . . . . . . . . 123\n4.22 Values of the relative leakage δQp/IGal;p for a pair of detectors . . . 123\n4.23 Estimated leakage variance of the Q component relative to the dust\n\ntemperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125\n4.24 Estimated variance distribution of the relative leakage parameter . . 126\n4.25 Effect of a poorly chosen scanning frequency ratios . . . . . . . . . 126\n4.26 The hit count map of a pair detector in a assumed pessimistic case 127\n4.27 The hit count map of a pair detector for the fiducial scanning strategy128\n4.28 A bad chosen parameter of the precession time . . . . . . . . . . . . 129\n4.29 Power spectrum of a badly chosen parameter of precession time . . 130\n4.30 Variation of precession time parameter . . . . . . . . . . . . . . . . 131\n\n\n\nList of Figures xi\n\n4.31 Variation ratio of precession time and spin parameters . . . . . . . 132\n4.32 Histogram of the covariance matrices of the configuration α = 30◦,\n\nβ = 65◦, τspin = 2 min, τprec = 4 day . . . . . . . . . . . . . . . . . . 134\n4.33 Histogram of the covariance matrices of the configuration α = 50◦,\n\nβ = 45◦, τspin = 2 min, τprec = 4 day . . . . . . . . . . . . . . . . . . 134\n4.34 Histogram of the covariance matrices of the configuration α = 50◦,\n\nβ = 45◦, τspin = 10 min, τprec = 96.1803 min . . . . . . . . . . . . . . 135\n4.35 Histogram of the covariance matrices of the configuration α = 65◦,\n\nβ = 30◦, τspin = 10 min, τprec = 96.1803 minutes . . . . . . . . . . . 135\n\n5.1 The first observation of superconductivity . . . . . . . . . . . . . . 139\n5.2 The BCS theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140\n5.3 The simple thermal model of a bolometer and a TES . . . . . . . . 142\n5.4 The TES block diagrams . . . . . . . . . . . . . . . . . . . . . . . . 143\n5.5 The theoretical TES circuit . . . . . . . . . . . . . . . . . . . . . . 143\n5.6 The plot of temperature versus resistance . . . . . . . . . . . . . . . 147\n5.7 The linear response of the current and the heat power . . . . . . . . 150\n5.8 Gometry parameters of the absorbing grid . . . . . . . . . . . . . . 155\n5.9 The critical temperature of a TES depending on the percentage of\n\nniobium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156\n5.10 The TES array mask . . . . . . . . . . . . . . . . . . . . . . . . . . 157\n5.11 TES fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158\n5.12 The 256 TESs array . . . . . . . . . . . . . . . . . . . . . . . . . . 159\n5.13 The measured resistor result of connection between NbSi and Al\n\nwires for array P90. . . . . . . . . . . . . . . . . . . . . . . . . . . . 161\n5.14 The vacuum chamber of Triton 200/400 . . . . . . . . . . . . . . . 163\n5.15 Readout electronic components . . . . . . . . . . . . . . . . . . . . 164\n5.16 The scheme of the TES readout system . . . . . . . . . . . . . . . . 164\n5.17 128:1 Time-Domain SQUID multiplexer . . . . . . . . . . . . . . . . 165\n5.18 The sampling rate of 128 TESs . . . . . . . . . . . . . . . . . . . . 166\n5.19 The QUBIC studio interfaces . . . . . . . . . . . . . . . . . . . . . 167\n5.20 The block diagram of a TES and the readout system . . . . . . . . 168\n5.21 Layout of TES array after the readout system . . . . . . . . . . . . 169\n5.22 A microphotography of a SQUID bonding . . . . . . . . . . . . . . 171\n5.23 SQUID read out system . . . . . . . . . . . . . . . . . . . . . . . . 172\n5.24 IV, VP, VR curves of TES . . . . . . . . . . . . . . . . . . . . . . . 174\n5.25 Calibrate on critical temperature signal . . . . . . . . . . . . . . . . 175\n5.26 Americium source . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177\n5.27 The stopping power of alpha particles in different material . . . . . 178\n5.28 The simple model of TESs array . . . . . . . . . . . . . . . . . . . . 179\n5.29 A single particle hits the pixel event . . . . . . . . . . . . . . . . . . 183\n5.30 The data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185\n5.31 Glitches data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 186\n5.32 The data histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . 188\n5.33 Histograms of glitches peaks . . . . . . . . . . . . . . . . . . . . . . 188\n\n\n\nList of Figures xii\n\n5.34 A glitches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189\n5.35 Example of fitted glitches for pixel 88 . . . . . . . . . . . . . . . . . 191\n5.36 Time constants and peak . . . . . . . . . . . . . . . . . . . . . . . . 192\n5.37 The stacking glitches and median glitches methods . . . . . . . . . 193\n5.38 the first population of the electrical time constant τ0 . . . . . . . . 193\n5.39 The second population of the electrical time constant τ0 . . . . . . . 194\n5.40 The fitted data for different pixels . . . . . . . . . . . . . . . . . . . 195\n5.41 The micro-photography of several pixels . . . . . . . . . . . . . . . 196\n5.42 The evidence of crosstalk . . . . . . . . . . . . . . . . . . . . . . . . 198\n5.43 The fitted model of time constants for the different KI parameters . 199\n5.44 Time constants respect to the KI parameter of the PID controller . 200\n5.45 The fitted model of time constants for the different voltage-bias VTES201\n5.46 Time constants, amplitude versus voltage bias VTES parameters . . 202\n5.47 The cross-talk analysis scheme . . . . . . . . . . . . . . . . . . . . . 204\n5.48 3.84 s TOD for pixels . . . . . . . . . . . . . . . . . . . . . . . . . . 204\n5.49 The histogram of cross-talk . . . . . . . . . . . . . . . . . . . . . . 207\n5.50 The 32x4 SQUIDs and the ASIC multiplexing readout . . . . . . . 209\n5.51 cross-talk of the electronic readout chain . . . . . . . . . . . . . . . 210\n\n\n\nList of Tables\n\n2.1 The solution of fluid equation and Friedmann equation . . . . . . . 23\n\n3.1 The cosmological parameters table . . . . . . . . . . . . . . . . . . 48\n3.2 Systematic table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78\n3.3 QUBIC experiment general information . . . . . . . . . . . . . . . . 85\n3.4 Detector configuration . . . . . . . . . . . . . . . . . . . . . . . . . 92\n\n4.1 Contribution of bandpass mismatch error to the r . . . . . . . . . . 114\n\n5.1 The table of angular resolution . . . . . . . . . . . . . . . . . . . . 156\n5.2 The table of TES arrays . . . . . . . . . . . . . . . . . . . . . . . . 162\n5.3 The table of the deposited energy . . . . . . . . . . . . . . . . . . . 178\n5.4 The table of materials . . . . . . . . . . . . . . . . . . . . . . . . . 182\n5.5 The tabke of measurement data . . . . . . . . . . . . . . . . . . . . 184\n5.6 The table of fitted parameters for run7 . . . . . . . . . . . . . . . . 194\n5.7 The table of time constants respects to KI parameters . . . . . . . 200\n5.8 The table of time constants respect to the voltage-biased VTES . . . 202\n5.9 The table of cross-talk estimators for different baseline positions . . 206\n5.10 The table of thermal cross-talk values . . . . . . . . . . . . . . . . . 208\n5.11 The table of contribution of noise level to thermal cross-talk estimator208\n\nxiii\n\n\n\nAbbreviations\n\nAPC Laboratoire AstroParticule et Cosmologie\n\nASIC Application Specific Integrated Circuit\n\nCMB Cosmic Microwave Background radiation\n\nCOBE COsmic Background Explorer\n\nCDM Cold Dark Matter\n\nCRs Cosmic Rays\n\nETF ElectroThermal Feedback\n\nFLL Flux Locked Loop\n\nFPGA Field Programmable Gate Array\n\nHFT High Frequency Telescope\n\nHWP Half Wave Plate\n\nISM Inter-Stellar Medium\n\nLFT Low Frequency Telescope\n\nLNA Low Noise Amplification\n\nNEP Noise Equivalent Power\n\nNET Noise Equivalent Temperature\n\nPID Proportional Integral Derivative\n\nPSD Power Spectral Density\n\nQUBIC Q and U Bolometric Interferometer fo Cosmology\n\nSQUID Superconducting Quantum Interference Device\n\nTDM Time Domain Multiplexing\n\nTES Transition Edge Sensor\n\nTOD Time Order Data\n\nWMAP Wilkinson Microwave Anisotropy Probe\n\nxiv\n\n\n\nChapter 1\n\nIntroduction\n\nSince the beginning of my Ph.D in November 2015, I have worked on 2 main topics:\n\n• Study of the bandpass mismatch systematic effect for future CMB projects\n(LiteBIRD, CORE).\n\n• Instrumentation: Behaviour of a TES array (QUBIC) and its electronic read-\nout system, subjected to particles interactions.\n\n1.1 Evaluating the level of the bandpass mismatch\n\nsystematic effect for the future CMB satellites\n\nThe future Cosmic Microwave Background (CMB) satellite concepts LiteBird [78],\nCORE [32], PIXIE [67] have been proposed to probe B mode polarization to mea-\nsure the tensor-to-scalar r ratio with a sensitivity σ(r) ≤ 10−3, which is almost\ntwo orders of magnitude beyond the Planck sensitivity. Several important system-\natic effects could contribute to final observation as 1/f noise, asymmetric beams,\nbandpass mismatches, the interaction of cosmic rays with the focal plane etc.\n\nWe learn from Planck mission data analysis that bandpass mismatch error is one\nof the important systematic effects that can affect the current and next-generation\nmeasurements of the polarization of the Cosmic Microwave Background radiation\n(CMB). The slightly different frequency bandpasses among detectors introduce\nleakage from intensity into CMB polarization. With the help of full focal plane\n\n1\n\n\n\nIntroduction 2\n\nsimulations, I evaluated the level of the bandpass mismatch systematic effect for\nfuture CMB satellites and estimated its possible impact on the final determination\nof the tensor-to-scalar ratio r. I simulated the time streams with filter variations\nas observed in Planck HFI. I assumed nominal scan and detector parameters for\nLiteBIRD. I projected data using the simplest map-making coaddition method.\nPower spectra of residual EE and BB coming from the leakage maps are computed\nfor 80 % sky fraction excluding the galactic plane. The amplitude of leakage\ndepends on the scanning strategy of the satellite parameterized with precession\nangle α, spin angle β, precession spin ωprec and rotating spin ωspin. I verified\nan analytic estimation which has shown the tight correlation between leakage\nmaps and the crossing moment, 〈cos 2ψ〉, 〈sin 2ψ〉, this is a fast and easy way to\npredict the magnitude of potential leakage. I found that the spurious signal could\npotentially bias r for measurements of the reionization bump (2 ≤ ` ≤ 10) at the\nlevel of about 5 × 10−4, and of the recombination bump (10 ≤ ` ≤ 200) at the\nlevel of about 5× 10−5 depending on scanning angle parameters. I demonstrated\nthe amplitude scales as the number of detectors. The effect is negligible in case of\nan ideal HWP. This study has led to a publication [50].\n\n1.2 Interaction of particles with a TES array\n\nIn a normal conductor/semiconductor the current is carried by electrons (i.e.\nfermions: half-integer spin) which obey Fermi-Dirac statistics while in a supercon-\nductor the current is carried by cooper-pairs (i.e. bosons: integer spin) which obey\nBose-Einstein statistics. The principle of superconductivity is that the phonon lat-\ntice slows down the velocity of electrons so that the electrons joined into cooper-\npairs. The development of a sensitive superconducting detector allows us to mea-\nsure a power source with faster responses and a larger heat capacity [61]. A\nsuperconducting detector works at a low temperature in that case the noise level\nis reduced closely to quantum limit. The superconducting transition regime is a\ntiny change in temperature, of the order of 0.1 mK to 1 mK leading to a large\nchange of resistance. For this reason a superconductor is an ideal candidate for a\nthermometer in a bolometer. When the thermometer is voltage biased, the Joule\npower is given by PJ = V2/RTES, a rise in temperature leads to an increased re-\nsistance of the thermometer, then the Joule power decreased. It means that the\nJoule power compensates the original rising temperature. This effect is known as\n\n\n\nIntroduction 3\n\nthe strong negative electrothermal feedback (ETF) which maintains the transition\nedge sensor (TES) temperature stable in the superconducting transition regime.\nThis physical characteristic is useful for an array of TESs and is one of the main\ninterest of the use of TES. Furthermore, the effective thermal time constant of\na transition edge sensor is divided by the loop gain L parameter of the ETF.\nTherefore the TES can be used as very sensitive and linear detectors operated\nsimultaneously in an array.\n\nI have studied the interaction of particles with a 256 TESs array of the ground-\nbased QUBIC (Q&U bolometric interferometer for cosmology) experiment [30].\nIn order to test the sensitivity of detectors to cosmic rays, an Americium 241 ra-\ndioactive source was set up in front of the 256 TES array in the mixing chamber\ninside the cryostat at 300 mK. When particles hit the components of a TES (eg:\nthermometer, absorbing grid, substrate), the deposited energy induced to temper-\nature elevation of the components and to the neighbor pixels, which provides a\nevaluation of the cross-talk. Moreover, this study allows us to understand both\nthe intrinsic time constant of a TES and the readout system time constant. First\nof all, I used a source of α particles from Americium 241 hitting the silicon wafer\nor the TES. In order to analyze time constants of a glitch, following the thermal\nresponse equation of a TES, I found thermal saturating equations of the wafer\nand the TES. By solving those equations I ended up with the time constants as\nexponential functions.\n\nOn the other hand, the electronic readout system of the TES has the following com-\nponents the connection of the TESs array, the 128 multiplexing superconducting\nquantum interference device (SQUIDs), the application specific integrated circuit\n(ASIC), and the warm digital readout. Applying the block diagram and automatic\ncontrol algorithm for the readout electronic components, I found that those time\nconstants are inversely proportional to the integral parameter of the proportional\nintegral derivative (PID) controller of the Flux Locked Loop (FLL). The analysis\nof the glitch timeline indicated that there are two-time constants [131], the ther-\nmal time constant of the TES due to the deposited energy by particles and the\nelectronic readout system time constant which depends on the FLL parameter.\nThe thermal cross-talk is constrained to be less than 0.1 percent. The low statis-\ntics of events do not allow to put a better constraint. In addition, the electronic\nreadout system can introduce the cross-talk between two successive pixels in the\nmultiplexing timeline.\n\n\n\nModern Cosmology 4\n\nFor pedagogical purposes, I write this thesis with many details, developed calcu-\nlations in the chapters 2 and 3. The chapter 4 is related to my simulation work\non bandpass mismatch systematic effect while the chapter 5 is my contribution in\ninstrumentation and data analysis. The manuscript is organized as follow:\n\nChapter 2 gives a quick introduction to modern cosmology. In this chapter, I\ndescribed the hot Big Bang Universe, the standard model of cosmology, the pri-\nmordial quantum fluctuations in inflation.\n\nChapter 3 provides the description of Cosmic Microwave Background and its future\nprojects which aim to detect the tensor-to-scalar ratio of the primordial fluctua-\ntions by measure B-mode polarization signal. The measurement of cosmic mi-\ncrowave background polarization has to cope with many challenges as foreground\nand systematic effects.\n\nChapter 4 details my study of bandpass mismatch error systematic effect for fu-\nture cosmic microwave background projects in particular the LiteBIRD satellite\nmission.\n\nChapter 5 details my study of the interaction of particles with the superconducting\ntransition edge sensors array of the ground-based QUBIC experiment.\n\nChapter 6 provides the conclusions of my thesis and perspectives after my thesis.\n\n\n\nChapter 2\n\nIntroduction to cosmology\n\nContents\n2.1 The Hot Big Bang theory . . . . . . . . . . . . . . . . . 6\n\n2.2 The standard cosmological model . . . . . . . . . . . . . 15\n\n2.3 Expanding Universe . . . . . . . . . . . . . . . . . . . . . 16\n\n2.4 General relativity . . . . . . . . . . . . . . . . . . . . . . 17\n\n2.4.1 Friedmann–Lemaître–Robertson–Walker (FLRW) metric 18\n\n2.4.2 Geodesic . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n\n2.4.3 Einstein equations and Friedmann equations . . . . . . . 20\n\n2.5 Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . 27\n\n2.6 The horizon problem . . . . . . . . . . . . . . . . . . . . 28\n\n2.7 The flatness problem . . . . . . . . . . . . . . . . . . . . 30\n\n2.8 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30\n\n2.9 Physics of inflation . . . . . . . . . . . . . . . . . . . . . . 32\n\n2.9.1 Slow-Roll inflation . . . . . . . . . . . . . . . . . . . . . 34\n\n2.10 Primordial quantum fluctuations in inflation and cos-\n\nmological perturbations . . . . . . . . . . . . . . . . . . . 36\n\n2.10.1 Linear perturbation . . . . . . . . . . . . . . . . . . . . 37\n\n2.10.2 Primordial quantum fluctuations in inflation . . . . . . . 38\n\n2.10.3 Cosmological perturbations and structure formation . . 43\n\n5\n\n\n\nThe Hot Big Bang theory 6\n\nSince the beginning of humankind, we were always curious about the dark and\nbright sky, the Solar system, the stars, the galaxies, and the Universe. We kept\nasking ourself what are we? where do we come from? and where are we going?\nThese questions are also represented by Paul Gauguin’s masterpiece as shown in\nfigure 2.1. I found that these questions apply to our Universe. The study of the\nUniverse provides scientific answers to our curiosities.\n\nFigure 2.1: D’où venons-nous? Que sommes-nous? Où allons-nous? Where Do\nWe Come From? What Are We? Where Are We Going?, 1897, Paul Gauguin,\nBoston Museum of Fine Arts. The Gauguin’s masterpiece should read from right\nto left, the baby means the beginning of life, where do we come from? in the\nmiddle of the painting is human density, ambition, desire, destiny, what are we\ndoing? On the left is a statue, it represents for spiritual belief which is beyond\nthe Earth, at the bottom left, the young woman reflected an old woman who is\n\ngoing to die, where are we going?\n\nIn this chapter, I describe the history of the Hot Big Bang theory, then our knowl-\nedge about the Universe by studying the standard cosmological model. I also\nfocus on inflation Universe theory which can solve the horizon problem (the ho-\nmogeneous Universe) and the flatness problem. I also present initial quantum\nfluctuations during inflation which are the seed of the large-scale structures in the\npresent complex Universe.\n\n2.1 The Hot Big Bang theory\n\nThe physical cosmology is the branch of the astronomy that deals with the origin\nand evolution of the Universe as a whole. In the 16th century, Nicolaus Copernicus,\n\n\n\nThe Hot Big Bang theory 7\n\na Polish scientist suggested that the Sun is center of the Solar system. In the 17th\n\ncentury, Isaac Newton solved the planetary motion by introducing the gravitation\nforce. The modern cosmology started in the 20th century, three century after Isaac\nNewton, in 1917, Albert Einstein published the theory of gravity in the paper\ncosmological considerations of the General Theory of Relativity [41]. In 1929,\nEdwin Hubble discovered the redshift of the light of distant galaxies, meaning\nthat they are rushing far away from the Milky Way with a velocity proportional\nto the distance so that the Universe must be expanding. Therefore the hot Big\nBang model which was suggested by Georges Lemaitre in 1927, was accepted.\nThe Big Bang theory described the Universe as a whole and have begun 13.798 ±\n0.037 billion years ago. The Universe contains 4.9 % ordinary matter, 26.8 % dark\nmatter and 68.3 % dark energy [119]. The evolution of the Universe is illustrated\nin figure 2.2 described by the Big Bang theory.\n\nScale a(t)\n\nTime [years]\n\nRedshift\n\nEnergy\n\nIn\nfla\n\ntio\nn\n\nBB\nN\n\nRec\nom\n\nbin\nati\n\non Dark\n ag\n\nes Reio\nniz\n\nati\non\n\nGalaxy fo\nrmation\n\nDark energy\n\n? neutrinos\n\nCosmic Microwave Background\n\nB-mode polarization\n\ndensity \nfluctuation\n\ngravity waves\n\nLensing\n\nBAO\nLSS\n\nQSO Ly\tα\n\nIa\n21 cm\n\nGW\n\n13.7 billion380 0003 min\n\n026251100\n\n1 meV1 eVGeV\n\n10�34s\n<latexit 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sha1_base64=\"/UqEI4yiHMH7v5sQ9WoiDxuipMo=\">AAAB7nicbVBNSwMxEJ2tX7V+VT16CRbBU0kE0WPRi8cK9gPatWTTbBuazS5JVihLf4QXD4p49fd489+YbfegrQ8GHu/NMDMvSKQwFuNvr7S2vrG5Vd6u7Ozu7R9UD4/aJk414y0Wy1h3A2q4FIq3rLCSdxPNaRRI3gkmt7nfeeLaiFg92GnC/YiOlAgFo9ZJHYIfM3I5G1RruI7nQKuEFKQGBZqD6ld/GLM04soySY3pEZxYP6PaCib5rNJPDU8om9AR7zmqaMSNn83PnaEzpwxRGGtXyqK5+nsio5Ex0yhwnRG1Y7Ps5eJ/Xi+14bWfCZWkliu2WBSmEtkY5b+jodCcWTl1hDIt3K2IjammzLqEKi4EsvzyKmlf1Amuk3tca9wUcZThBE7hHAhcQQPuoAktYDCBZ3iFNy/xXrx372PRWvKKmWP4A+/zB0ZVjtk=</latexit>\n\nFigure 2.2: The history and the evolution of the Universe in time and scale\nfactor. Acronyms BBN: Big Bang Nucleosynthesis, Lyα: Lyman alpha, QSO:\nQuasi-Stellar Objects = Quasars, BAO: Baryon Acoustic Oscillation, LSS:\nLarge-Scale Structure, GW: Gravitational waves, 21cm: Hydrogen transition\n\nline and 21 cm cosmology, Ia: Type Ia supernovae\n\nThe thermal history of the early Universe is well described by the laws of particle\n\n\n\nThe Hot Big Bang theory 8\n\nphysics. Figure 2.3 mainly illustrates the evolution of the Universe at the beginning\nof time to 3 minutes. The events are summarized in the following:\n\nFigure 2.3: The early the Universe from the initial singularity to cosmic infla-\ntion, quantum gravity epoch, nucleosynthesis, last scattering surface [147].\n\nInflation epoch. t ∼ 10−34 s. In 1979 and 1980 Alain Guth proposed a theory\nof the exponential expansion of space-time in the early Universe to solve possibly\nthe horizon and flatness problems. The theory is developed by contributions of\nAndrei Linde, Paul Steinhardt and many physicists who believe in the inflation\nscenario of the Universe.\n\nBaryogenesis. t < 10−10 s. If the beginning Universe has equal amount of particle\nand antiparticle. Then the annihilated process of particles leads to a radiation\nUniverse. However, we observed today that the matter is dominated anti-matter\nin the present Universe. The hypothesis is that there was a time in the primordial\nUniverse when the symmetry was broken by some dynamic mechanisms which are\nnot known now. This issue is a puzzle in the modern cosmology. The observed\nratio of baryons to photons is:\n\nη ≡ nb − nb̄\n\nnγ\n∼ 10−9. (2.1)\n\nIn 1928, Paul Dirac published the paper \"The quantum theory of the electron\"\n[4]. Dirac equation predicted the existence of antimatter. Five years later, in 1932\nCarl D. Anderson discovered the positron experimentally since then, every known\nkind of particle has an anti-particle. Particle and anti-particle have the same mass\nand opposite charge (e.g., electron and positron), together they annihilate. It is\ncalled C-symmetry.\n\n\n\nThe Hot Big Bang theory 9\n\nBaryon number is calculated by number of quarks and antiquarks B = (nq−nq̄)/3.\nBaryons made of 3 quarks then baryon number equals +1. Antibaryons made of 3\nantiquarks then baryon number equals -1. The thermal equilibrium of the Universe\nshould produce the same amount of matter and antimatter. However we observe\nin the present Universe the domination of matter, the abundance of anti-matter\nis very small. This problem is called baryon asymmetry in the Big Bang model.\nIn 1967, Andrei Sakharov indicated three necessary conditions for creating baryon\nasymmetry in the Universe.\n\n• Violation of Baryon number B\n\n• Violation of C-symmetry and CP-symmetry\n\n• State out of thermal equilibrium\n\nCP-symmetry violation is the violation of charge-parity symmetry or the combina-\ntion of C-symmetry and P-symmetry. P-symmetry is the change of sign (mirror)\nin parity transformation. The third condition is that in the state out of thermal\nequilibrium, the pair-annihilation is decreased then the particles and anti-particles\ndo not reach the equilibrium state.\n\nThe Large Hadron Collider (LHCb) accelerator can reach to the energy of the order\n1013 eV, which corresponds to the quark era of the early Universe. The particle\nphysics, the theory of general relativity and several hypotheses exist to explain\nthe problem, nevertheless the origin of matter-antimatter remains an unsolved\nproblem.\n\nElectroweak phase transition. When the temperature of the Universe was\naround 100 GeV, at t ∼ 10−10 s, particles (Z and W± Gauge bosons) interacted\nthrough the Higgs mechanism. Leptons (electron e, muon µ, tau τ , electron neu-\ntrino νe, muon neutrino νµ, tau neutrino ντ ) and quarks (up u, charm c, top t,\ndown d, strange s, bottom b) are in thermal equilibrium above 100 GeV. There\nis a Possible link between the electroweak phase transition and the dark matter\nof the Universe. If dark matter is made of particles, the dark matter particles\ninteract via gravity and might be weakly interacting massive particles (WIMP).\nThe WIMP model is still an hypothesis and many experiments attempt to detect\nthose. The symmetry between the weak forces and the electromagnetic is broken\nbelow 100 GeV.\n\n\n\nThe Hot Big Bang theory 10\n\nQCD phase transition. At the temperature of 100 - 300 MeV, t ∼ 10−5 s,\nbaryons appeared due to the strong interaction between quarks and gluons g (a\nGauge bosson). Three quarks systems are baryons as protons, neutrons, and\nquark-antiquark systems are mesons. There is no observed evidence of this phase\ntransition.\n\nNeutrino decoupling. At t ≈ 1 s, Energy ∼ 1 MeV. It occurred within one\nsecond. The weak interaction between neutrinos and the plasma matter produced\na cosmic neutrino background (CNB) which freely propagate into the Universe.\nThis epoch has happened before the recombination.\n\nElectron-positron annihilation. At t ≈ 6 s, Energy ∼ 500 keV, e−+e+ → γ+γ.\nThe energy of the electrons and positrons is transferred to the photons energy, and\nnot to neutrinos. It explains why the temperature of CNB is lower than the CMB\n(Tν < Tγ).\n\nBig Bang Nucleosynthesis (BBN). At t ≈ 3 minutes, Energy ∼ 0.1 MeV,\nredshift z ≈ 4× 108. The percentage of the light elements - deuterium, helium 3,\nlithium, helium 4 - in the Universe is an evidence supporting the Hot Big Bang\ntheory. At high temperature, protons and neutrons stay in a thermal equilibrium\nstate. The Universe is expanding and cools down enough, then these particles\nbind into nuclei. The production of light elements is processed through a series\nreaction chain.\n\np + n→ D;\n\nD + p→ 3He;\n\nD + D→ 4He;\n\n. . .\n\nHere D stands for deuterium nuclei. These processes happen when the age of the\nUniverse is ∼ 3 minutes. [12] Figure 2.4 shows the observational data is agreement\nwith the predicted abundance of light elements. It means that the result is a good\nevidence of light elements being formed in the Big Bang Nucleosynthesis theory.\nThe abundance of 4He depends on the abundance of neutrous which started to de-\ncay at the time of nucleosynthesis. The heavier elements are made in stars, during\nstellar evolution or in supernovae. This has originally been described in 1948, R.\nA. Alpher, H. Bethe and G. Gamow who predicted elements in the Universe by a\npublished paper is The Origin of Chemical Elements (αβγ paper) [12].\n\n\n\nThe Hot Big Bang theory 11\n\nFigure 2.4: The primordial predicted abundance of light elements in the Uni-\nverse on the density of ordinary matter (∼ 4%) and the WMAP satellite.Credit:\n\nWMAP/NASA team\n\nRecombination. At t ≈ 380000 years, Energy ∼ 0.1 eV, z ≈ 1100. When the\ntemperature cooled down to T ∼ 3000 K, the electrons combined with protons\nand formed neutral Hydrogen through the process: e− + p+ → H + γ. This is the\nfirst phase change of the Universe with the formation of neutral hydrogen atoms.\n\nPhoton decoupling. At t ≈ 400000 years, Energy ∼ 0.1 eV, z ≈ 1100. The\nphotons interact with the electrons through Thomson scattering e− + γ → e− +\n\nγ, those polarized photons (photon decouple) freely traveled entire Universe and\nwe observed today as the Cosmic Microwave Background (CMB) polarization.\nThe fraction of free electrons decrease suddenly because of recombination. I will\ndescribe detail the CMB physics in the chapter 3.\n\nThe Epoch of Reionization (EoR). At t ≈ 108 years, Energy ∼ 1 meV, z ≈ 6-\n25. After the recombination epoch, the Universe is composed of neutral Hydrogen.\n\n\n\nThe Hot Big Bang theory 12\n\nThe Hydrogen atoms concentrate to form gas clouds due to gravitation. First\nstars are formed in the interstellar medium (ISM), however, the UV radiation\nfrom first stars reionized hydrogen in the ISM. The Universe changed from neutral\nto ionize state. The 21 cm hydrogen emission line occurred due to the energy\nlevel of electron and proton in a hydrogen atom. Observing 21 cm radiation is a\nuseful tool to study the reionization epoch and the properties of the ISM, and the\ntopology (the local and global geometry) of the Universe. The Lyman α forest at\nredshift z ≈ 2.5-6.5, allows to probe the state of the intergalactic medium (IGM)\nvia the absorption of a neutral hydrogen in the spectra of a distant quasi-stellar\nobjects (QSO) 1 when the Universe was filled with gas.\n\nFigure 2.5: The plot shows the observed magnitude data of Type Ia supernovae\nfor different experiments versus redshift, it implies that the expanding Universe\n\nis acceleration. [95]\n\nDark energy epoch. At t ≈ 109 years, Energy ∼ 1 meV, z ≈ 0-2. At this epoch,\nthe Universe is dominated by the dark energy and the expansion is accelerating.\nWe usually assume that the dark energy has negative pressure and distributes\nhomogeneously in space. The accelerating Universe was first evidenced by the\nobservation type Ia supernovae. Supernova is an explosion of a massive star 2 in the\nuniverse. Type Ia supernovae result from stars accumulating matter from nearby\nneighbor stars and collapsing together with a white dwarf. Type Ia supernovae\nhave similar masses, then their luminosity have the same brightness. They are\n1It is also known as Quasar. A Quasar is an extremely luminous distant object. Quasars energy\nis believably powered from the accretion disk of massive black holes at the center of an active\ngalactic nucleus (AGN).\n\n2Type Ia occurs in binary systems in which two stars orbit each other.\n\n\n\nThe Hot Big Bang theory 13\n\nconsidered as a standard candle in the Universe. The dynamics of the Universe\ncan be inferred by measuring luminosity distance of objects then comparing to their\ncorresponded redshift of the standard candle. The result indicated the accelerating\nUniverse as shown in figure 2.5.\n\nPresent and observation. Today, we are in the dark energy dominated epoch.\nOur Universe is ∼ 13.7 million years and contains ∼ 4.9 % of ordinary matter, ∼\n26.8 % of dark matter and ∼ 68.3 % of dark energy (Planck result).\n\nFigure 2.6: The 2dF galaxy redshift survey data release. The Earth is placed at\nthe center. The white part is the un-observed sky. Each data point is equivalent\nto a galaxy. The 2dF galaxy survey observed about hundred thousand galaxies.\n\nCredit: 2dFGRS team\n\nBy surveying the Large-Scale Structure (LSS) of galaxies clusters using redshift\nwhich is hundreds of Mpc and more. The 2dF galaxy survey (figure 2.6) and Sloan\nDigital Sky Survey (SDSS) surveyed typically about hundred thousands or billions\nof galaxies in the Universe. The smoothness of the Universe is a fundamental\nassumption in cosmology. The Baryon Acoustic Oscillation (BAO) are acoustic\noscillations observed on galaxy-galaxy correlations that can be observed in the\nlarge-scale structure of galaxies in the Universe. BAO is a standard ruler for the\nlength scale in the Universe. In 2014, the SDSS’s Baryon Spectroscopic Survey at\nredshift 0.2 < z < 0.7 detected the BAO signal with 7σ [13].\n\n\n\nThe Hot Big Bang theory 14\n\nFigure 2.7: In 2016, LIGO collaboration discovered the first detection of grav-\nitational waves by observing two black holes merger [11].\n\nIn 2016, the LIGO/VIRGO collaboration discovered the existence of gravitational\nwaves which open a new window on the Universe. Gravitational waves are gener-\nated by merged massive objects. The existance of gravitational waves was proposed\nby Henri Poincaré based on the theory of relativity. Gravitational waves travel\nat the speed of light and transform energy as electromagnetic energy. The first\ndetection results from the merging of a binary black hole with masses 36 Msun and\n29 Msun [11] and the second detection concerns the binary neutron star inspiral.\n\nNeutrinos. There have three kinds of neutrino: electron neutrino νe, muon neu-\ntrino νµ, tau neutrino ντ . Neutrino oscillations are the phenomenon of changing\ntheir types when they travel, for example, an electron neutrino can become a\nmuon neutrino then oscillating back to its original type. Neutrinos are believed\nto have non-zero rest mass, and they have very weak interaction with matter\nin the Universe. Along with the cosmic neutrinos background which have not\nbeen detected directly but though the effect on CMB at the early Universe, there\nhave some sources of neutrinos directly detected: Our Sun, nearby supernova\nin 1987, recently the multiteam collaboration-the IceCube Collaboration, Fermi-\nLAT, MAGIC, AGILE, ASAS-SN, HAWC, H.E.S.S., INTEGRAL, Kanata, Kiso,\n\n\n\nThe standard cosmological model 15\n\nKapteyn, Liverpool Telescope, Subaru, Swift/NuSTAR, VERITAS, VLA/17B-403\nteams- has been discovered a high energy neutrino astrophysical source from a\nblazar flare which is a type of quasar. Blazars also open a new window on mul-\ntimessenger astronomy for observations of cosmic rays, neutrinos, gravitational\nwaves, and electromagnetic [10].\n\nSince the Hot Big Bang theory has much observational evidence, it is true to state\nthat all of us made up of particles. The behavior of our Universe today depends\non the properties of particles at the early Universe.\n\n2.2 The standard cosmological model\n\nThe standard cosmological model considers the Universe as a whole and encom-\npasses our knowledge of it.\n\nGeometry, StructureDynamics\nGR, fluid eq., \n\n\nStress-Energy Tensor\nHomogeneous and isotropy, \n\n\nMetric \n\nRµ⌫ � 1\n\n2\ngµ⌫R � ⇤gµ⌫ = 8⇡GTµ⌫\n\n<latexit sha1_base64=\"vG3ryck1tC6aJwc9vTClH3lhQZI=\">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</latexit><latexit 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sha1_base64=\"3DsIRCYu2Htu8pA6HBebEigMs5M=\">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</latexit>\n\nds2 = �c2dt2 + a2(t)\n\n✓\ndr2\n\n1 � kr2\n+ r2\n\n�\nd✓2 + sin2 ✓d�2\n\n�◆\n\n<latexit sha1_base64=\"o0oKG3SWjgZS7p2O4AfuFVZmQf0=\">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</latexit><latexit sha1_base64=\"9rP5c3C6vSqwlZNnKNXjD2RFZSk=\">AAACa3icbVHBbtNAEF2bAmmAEuihCDiMqCqlqojsXMoFqYILx1YibaU4jdbrdbzKem3tjisFKxd+gCtfwAdx4w+49B86dlKptIy0O09v3szuvo1LrRwGwR/Pf7Dx8NHjzmb3ydNnW897L16euqKyQo5EoQt7HnMntTJyhAq1PC+t5Hms5Vk8/9zUzy6ldaowX3FRyknOZ0alSnAkatr7nriLIXyE9yAoJ0jbAXDa+7gPEGmZYh+iJLVc1Im9GC7rkLTzBpGQ0o0mgQgziRzaCZFTpkFrjoplphoCIqtmWTt7BbrdaW83GARtwH0QrsHu0f6Pb2Xn18/jae93lBSiyqVBoblz4zAocVJzi0pouexGlZMlF3M+k2OChufSTerWqyXsEZNAWlhaBqFlb3fUPHdukcekzDlm7m6tIf9XG1eYfpjUypQVSiNWB6WVBiygMR4SZaVAvSDAhVV0VxAZJ1+RvqcxIbz75PvgdDgIg0F4Qm58YqvosDfsHeuzkB2yI/aFHbMRE+yvt+XteK+8K3/bf+2/XUl9b92zzf4Jf+8aZ0G0ww==</latexit><latexit sha1_base64=\"9rP5c3C6vSqwlZNnKNXjD2RFZSk=\">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</latexit><latexit sha1_base64=\"9rP5c3C6vSqwlZNnKNXjD2RFZSk=\">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</latexit><latexit sha1_base64=\"O2I1v0mBBQtHEO9FYa7z4N06gRU=\">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</latexit>\n\nContents Early Universe\nMatter, radiation, \n\n\ncosmological constant\n\n⌦ = ⌦m + ⌦r + ⌦⇤\n<latexit sha1_base64=\"xa/fVAmTlLHum9BrW4ZSlzBw664=\">AAACHHicbZDLSgMxFIbPeK31Niq4cRMsgiCUGV3oRii6cSHYgr1Ap9RMmrahycyQZIQy9EHc+CpuXCjixoXgxmcx0xaprT8EPv5zTpLz+xFnSjvOlzU3v7C4tJxZya6urW9s2lvbFRXGktAyCXkoaz5WlLOAljXTnNYiSbHwOa36vcu0Xr2nUrEwuNX9iDYE7gSszQjWxmraJ96NoB2MztEYmgId/bKcYORdm2tbGGWbds7JO0OhWXDHkCvslr7vAKDYtD+8VkhiQQNNOFaq7jqRbiRYakY4HWS9WNEIkx7u0LrBAAuqGslwuQE6ME4LtUNpTqDR0J2cSLBQqi980ymw7qrpWmr+V6vHun3WSFgQxZoGZPRQO+ZIhyhNCrWYpETzvgFMJDN/RaSLJSba5JmG4E6vPAuV47zr5N2SSeMCRsrAHuzDIbhwCgW4giKUgcADPMELvFqP1rP1Zr2PWues8cwO/JH1+QPHd6A0</latexit><latexit sha1_base64=\"5UfQnB6z9mCN2iG7KXJ3fL0BmkQ=\">AAACHHicbZDLSgMxFIYzXmu9jQpu3ASLIAhlRhe6EUrduBBswV6gMwyZNNOGJjNDkhHK0Adx4xv4DG66UMSNC8GNz2KmLVJbfwh8/OecJOf3Y0alsqwvY2FxaXllNbeWX9/Y3No2d3brMkoEJjUcsUg0fSQJoyGpKaoYacaCIO4z0vB7V1m9cU+EpFF4p/oxcTnqhDSgGClteeaZc8tJB8FLOAGPw5NfFlMMnRt9bRvBvGcWrKI1EpwHewKF0n71mz6VhxXP/HDaEU44CRVmSMqWbcXKTZFQFDMyyDuJJDHCPdQhLY0h4kS66Wi5ATzSThsGkdAnVHDkTk+kiEvZ577u5Eh15WwtM/+rtRIVXLgpDeNEkRCPHwoSBlUEs6RgmwqCFetrQFhQ/VeIu0ggrHSeWQj27MrzUD8t2lbRruo0ymCsHDgAh+AY2OAclMA1qIAawOABPIMX8Go8GkPjzXgfty4Yk5k98EfG5w8Yx6Hw</latexit><latexit sha1_base64=\"5UfQnB6z9mCN2iG7KXJ3fL0BmkQ=\">AAACHHicbZDLSgMxFIYzXmu9jQpu3ASLIAhlRhe6EUrduBBswV6gMwyZNNOGJjNDkhHK0Adx4xv4DG66UMSNC8GNz2KmLVJbfwh8/OecJOf3Y0alsqwvY2FxaXllNbeWX9/Y3No2d3brMkoEJjUcsUg0fSQJoyGpKaoYacaCIO4z0vB7V1m9cU+EpFF4p/oxcTnqhDSgGClteeaZc8tJB8FLOAGPw5NfFlMMnRt9bRvBvGcWrKI1EpwHewKF0n71mz6VhxXP/HDaEU44CRVmSMqWbcXKTZFQFDMyyDuJJDHCPdQhLY0h4kS66Wi5ATzSThsGkdAnVHDkTk+kiEvZ577u5Eh15WwtM/+rtRIVXLgpDeNEkRCPHwoSBlUEs6RgmwqCFetrQFhQ/VeIu0ggrHSeWQj27MrzUD8t2lbRruo0ymCsHDgAh+AY2OAclMA1qIAawOABPIMX8Go8GkPjzXgfty4Yk5k98EfG5w8Yx6Hw</latexit><latexit sha1_base64=\"5UfQnB6z9mCN2iG7KXJ3fL0BmkQ=\">AAACHHicbZDLSgMxFIYzXmu9jQpu3ASLIAhlRhe6EUrduBBswV6gMwyZNNOGJjNDkhHK0Adx4xv4DG66UMSNC8GNz2KmLVJbfwh8/OecJOf3Y0alsqwvY2FxaXllNbeWX9/Y3No2d3brMkoEJjUcsUg0fSQJoyGpKaoYacaCIO4z0vB7V1m9cU+EpFF4p/oxcTnqhDSgGClteeaZc8tJB8FLOAGPw5NfFlMMnRt9bRvBvGcWrKI1EpwHewKF0n71mz6VhxXP/HDaEU44CRVmSMqWbcXKTZFQFDMyyDuJJDHCPdQhLY0h4kS66Wi5ATzSThsGkdAnVHDkTk+kiEvZ577u5Eh15WwtM/+rtRIVXLgpDeNEkRCPHwoSBlUEs6RgmwqCFetrQFhQ/VeIu0ggrHSeWQj27MrzUD8t2lbRruo0ymCsHDgAh+AY2OAclMA1qIAawOABPIMX8Go8GkPjzXgfty4Yk5k98EfG5w8Yx6Hw</latexit><latexit sha1_base64=\"NgJup6B4WPRoYojbhJtFVe8VrlE=\">AAACHHicbZBNS8MwGMdTX+d8q3r0EhyCIIxWD3oRhl48CE5wL7CWkqbZFpakJUmFUfZBvPhVvHhQxIsHwW9juhWZm38I/Pg/z5Pk+YcJo0o7zre1sLi0vLJaWiuvb2xubds7u00VpxKTBo5ZLNshUoRRQRqaakbaiSSIh4y0wsFVXm89EKloLO71MCE+Rz1BuxQjbazAPvVuOekheAELCDg8/mU5xdC7MddGCJYDu+JUnbHgPLgFVEChemB/elGMU06Exgwp1XGdRPsZkppiRkZlL1UkQXiAeqRjUCBOlJ+NlxvBQ+NEsBtLc4SGY3d6IkNcqSEPTSdHuq9ma7n5X62T6u65n1GRpJoIPHmomzKoY5gnBSMqCdZsaABhSc1fIe4jibA2eeYhuLMrz0PzpOo6VffOqdQuizhKYB8cgCPggjNQA9egDhoAg0fwDF7Bm/VkvVjv1sekdcEqZvbAH1lfPzjfnk8=</latexit>\n\nInflation, BBN, CMB\n\nExpansion, \na, H, h, z\n\nAge Fate\n\nt0 =\n2\n\n3\nH�1\n\n0\n\n= 6.51h�1 ⇥ 109 yrs\n<latexit sha1_base64=\"8eEj2M9NJdqijZKCX0yrmhdVjk8=\">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</latexit><latexit sha1_base64=\"8eEj2M9NJdqijZKCX0yrmhdVjk8=\">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</latexit><latexit sha1_base64=\"8eEj2M9NJdqijZKCX0yrmhdVjk8=\">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</latexit><latexit sha1_base64=\"8eEj2M9NJdqijZKCX0yrmhdVjk8=\">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</latexit><latexit sha1_base64=\"8eEj2M9NJdqijZKCX0yrmhdVjk8=\">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</latexit>\n\nH, z H, z, k, a\n\n⌦� 1 = � kc2\n\nH2a2\n<latexit 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sha1_base64=\"TUI3MXbHdGGCRLm65kV5MqrpmjI=\">AAACFHicbVC7SgNBFJ2NryS+opY2F4MgiGE3jTZC0CadEcwDs0mYncwmQ2YfzMyKYdmPsBH8CksbC0VsLez8Gp08Ck08MJfDOfdy5x4n5Ewq0/wyUguLS8sr6Ux2dW19YzO3tV2TQSQIrZKAB6LhYEk582lVMcVpIxQUew6ndWdwPvLrN1RIFvhXahjSlod7PnMZwUpLndwh2Bce7WGAI7DgVFe76wpM4gGQdjGJy9AuAtYlgWwnlzcL5hgwT6wpyZcy4cP14+13pZP7tLsBiTzqK8KxlE3LDFUrxkIxwmmStSNJQ0wGuEebmvrYo7IVj49KYF8rXXADoZ+vYKz+noixJ+XQc3Snh1Vfznoj8T+vGSn3pBUzP4wU9clkkRtxUAGMEoIuE5QoPtQEE8H0X4H0sc5E6RxHIVizJ8+TWrFgmQXrUqdxhiZIo120hw6QhY5RCZVRBVURQXfoCb2gV+PeeDbejPdJa8qYzuygPzA+fgAc5p4x</latexit><latexit sha1_base64=\"bWue5WLECHopB2FQn3A8CVhN9QQ=\">AAACFHicbVC7SgNBFJ2Nrxhfq5Y2F4MgiGE3jTZC0CadEcwDkk2YnZ1NhszOLjOzQljyETb+io2FIrYWdv6Nk0ehiQfmcjjnXu7c4yecKe0431ZuZXVtfSO/Wdja3tnds/cPGipOJaF1EvNYtnysKGeC1jXTnLYSSXHkc9r0hzcTv/lApWKxuNejhHoR7gsWMoK1kXr2GXRuI9rHAOfgwpWpnSCUmGRDIN3yOKtCtwzYlDEUenbRKTlTwDJx56SI5qj17K9OEJM0okITjpVqu06ivQxLzQin40InVTTBZIj7tG2owBFVXjY9agwnRgkgjKV5QsNU/T2R4UipUeSbzgjrgVr0JuJ/XjvV4aWXMZGkmgoyWxSmHHQMk4QgYJISzUeGYCKZ+SuQATaZaJPjJAR38eRl0iiXXKfk3jnFyvU8jjw6QsfoFLnoAlVQFdVQHRH0iJ7RK3qznqwX6936mLXmrPnMIfoD6/MH/GSaXw==</latexit>\n\nv = H0D\n<latexit sha1_base64=\"MSPoWJHImKU0E9+MqTZcB85mpvk=\">AAAB8HicbVC7SgNBFL0bXzG+ooKNzWAQrMKuTWyEoBYpEzAPSZY4O5lNhszMLjOzgbDkK2wsFLH1c+xs/BYnj0ITD1w4nHMv994TxJxp47pfTmZtfWNzK7ud29nd2z/IHx41dJQoQusk4pFqBVhTziStG2Y4bcWKYhFw2gyGt1O/OaJKs0jem3FMfYH7koWMYGOlhxG6RpWui+66+YJbdGdAq8RbkEL5pPb9CADVbv6z04tIIqg0hGOt254bGz/FyjDC6STXSTSNMRniPm1bKrGg2k9nB0/QuVV6KIyULWnQTP09kWKh9VgEtlNgM9DL3lT8z2snJrzyUybjxFBJ5ovChCMToen3qMcUJYaPLcFEMXsrIgOsMDE2o5wNwVt+eZU0LoueW/RqNo0bmCMLp3AGF+BBCcpQgSrUgYCAJ3iBV0c5z86b8z5vzTiLmWP4A+fjBzxnkOM=</latexit><latexit 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sha1_base64=\"985hjqRt7BgBJF8sdqz/EParfqQ=\">AAAB8HicbVC7SgNBFL3rM8ZXVLCxGQyCVdi10UYI0SJlAuYhyRJmJ7PJkJnZZWY2EJZ8hY2FIraCf+EX2Nn4LU4ehSYeuHA4517uvSeIOdPGdb+cldW19Y3NzFZ2e2d3bz93cFjXUaIIrZGIR6oZYE05k7RmmOG0GSuKRcBpIxjcTPzGkCrNInlnRjH1Be5JFjKCjZXuh+galTsuuu3k8m7BnQItE29O8sXj6jd7L31UOrnPdjciiaDSEI61bnlubPwUK8MIp+NsO9E0xmSAe7RlqcSCaj+dHjxGZ1bpojBStqRBU/X3RIqF1iMR2E6BTV8vehPxP6+VmPDKT5mME0MlmS0KE45MhCbfoy5TlBg+sgQTxeytiPSxwsTYjLI2BG/x5WVSvyh4bsGr2jRKMEMGTuAUzsGDSyhCGSpQAwICHuAJnh3lPDovzuusdcWZzxzBHzhvP42okp8=</latexit><latexit sha1_base64=\"ZNs3iFVrTEr4xjid1s8T/KfJfw8=\">AAAB8HicbVBNSwMxEJ2tX7V+VT16CRbBU8l60YtQ1EOPFWyttEvJptk2NMkuSbZQlv4KLx4U8erP8ea/MW33oK0PBh7vzTAzL0wENxbjb6+wtr6xuVXcLu3s7u0flA+PWiZONWVNGotYt0NimOCKNS23grUTzYgMBXsMR7cz/3HMtOGxerCThAWSDBSPOCXWSU9jdI3qPYzueuUKruI50Crxc1KBHI1e+avbj2kqmbJUEGM6Pk5skBFtORVsWuqmhiWEjsiAdRxVRDITZPODp+jMKX0UxdqVsmiu/p7IiDRmIkPXKYkdmmVvJv7ndVIbXQUZV0lqmaKLRVEqkI3R7HvU55pRKyaOEKq5uxXRIdGEWpdRyYXgL7+8SloXVR9X/Xtcqd3kcRThBE7hHHy4hBrUoQFNoCDhGV7hzdPei/fufSxaC14+cwx/4H3+AK3Ajv4=</latexit>\n\nFigure 2.8: The standard cosmological model included the accelerating ex-\npansion of the Universe, the age, the fate, the contents, the early Universe, the\ndynamic and geometry. a: scale factor. z: redshift. H, h, H: Hubble parameter,\n\nconstant. k: curvature Ω density parameters...\n\n\n\nExpanding Universe 16\n\nFigure 2.8 briefly summaries the standard cosmological model. Einstein intro-\nduced the cosmological constant as a term in the general relativity theory to\nexplain a static Universe, however observational data of Hubble showed that the\nUniverse is expanding. The results of supernovae Ia study and also CMB mea-\nsurements provided evidence of accelerating expansion and then of the presence\nof a cosmological constant. The cosmic microwave background observation helps\nus understand many aspects of the Universe including the geometry, the age, and\nthe contents. The early Universe requires more observation to test the inflation\ntheory.\n\n2.3 Expanding Universe\n\nIn 1929, at the Carnegie Observatories in Pasadena, California, Edwin Hubble\ndiscovered that the galaxies are moving away from us and the velocity of galaxy\nrecession is proportional to their distance from us so the light from distant galaxies\nwas redshifted [58]. This is an evidence of the expansion of the Universe. The\nso-called Hubble law express a linear relationship the recessional velocity and\ndistance.\n\nv = H0D. (2.2)\n\nHere v is the recessional velocity of distance objects km/s, H0 is the Hubble’s\nconstant or the Hubble parameter. D is the distance from the object to the\nobserver. The value of the Hubble constant is measured by many experiments as\nshown in figure 2.9. The latest Planck 2016 cosmological constraints since [110].\n\nH0 = 67.8± 0.9 kms−1Mpc−1.\n\nIt is useful to parameterize the Hubble constant as\n\nH0 = h · 100 kms−1Mpc−1. (2.3)\n\nThus h = 0.678±0.009 , assuming the Planck experimental measurement. Because\nof the expansion of the Universe, we can measure velocities of moving away objects\nusing the redshift defined as:\n\nz =\nλo − λe\n\nλe\n\n. (2.4)\n\n\n\nGeneral relativity 17\n\nFigure 2.9: The Hubble constant from the original paper and the measure-\nments by other experiments from 2001-2018\n\nsource: https://en.wikipedia.org/wiki/Hubble%27s_law.\n\nwhere λo, λe are the wavelength of the observation and emission, respectively. The\nscale factor is defined as the distance between two co-moving objects. The redshift\ncan be linked to the scale factor of the expanding Universe.\n\n1 + z =\nλo\nλe\n\n=\na(t0)\n\na(te)\n. (2.5)\n\n2.4 General relativity\n\nThe dynamics of the Universe is described by the general relativity. First of\nall, let us consider the distance between two events in four-dimensional space-time\n\n\n\nGeneral relativity 18\n\nand is invariant under coordinate transformations. It can be written as\n\nds2 =\n∑\n\nµ,ν\n\ngµνdx\nµdxν . (2.6)\n\nhere gµν is the metric tensor, µ, ν are index values 0, 1, 2, 3. x0 is time coordinate\nand the other are the three spatial coordinates. Notice that, In Einstein nota-\ntions, the lower indices indicate covariance (the perpendicular projections on the\ncoordinate axes) tensors, the upper indices indicate contravariance (the parallel\nprojections on the coordinate axes) tensors. In special relativity, the space time is\nthe Minkowski metric.\n\ngµν ≡ ηµν =\n\n\n\n\n1 0 0 0\n\n0 −1 0 0\n\n0 0 −1 0\n\n0 0 0 −1\n\n\n\n. (2.7)\n\nIn polar coordinates,\n\ngµν =\n\n\n\n\n1 0 0 0\n\n0 −1 0 0\n\n0 0 −r2 0\n\n0 0 0 −r2 sin2 θ\n\n\n\n. (2.8)\n\n2.4.1 Friedmann–Lemaître–Robertson–Walker (FLRW) met-\n\nric\n\nThe structure and evolution of the Universe are described by the Friedmann-\nRobertson-Walker (FRW) metric. It can be derived the Universe assuming homo-\ngeneity (same at every point) and isotropy (same at every direction) properties of\nthe Universe. The FRW metric for the spacetime writes:\n\nds2 = c2dt2 − a2(t)\n\n(\ndr2\n\n1− kr2\n+ r2\n\n(\ndθ2 + sin2 θdϕ2\n\n))\n, (2.9)\n\nwhere a(t) is the scale factor, r is a time dependent comoving coordinate, k is the\ncurvature parameter, k = +1 positive means an open Universe, k = -1 negative\nmeans a close Universe, k = 0 means a flat Universe, r, θ, ϕ are the spherical\ncoordinates.\n\n\n\nGeneral relativity 19\n\nIt is useful to express metric 2.9 in the comoving coordinates (hypersphere). This\nform is convenient for studying the propagation of light in the metric:\n\nds2 = dt2 − a(t)2\n(\ndχ2 + S2\n\nk(χ)\n(\ndθ2 + sin2 θdϕ2\n\n))\n, (2.10)\n\nwhere\n\nr2 = Sk(χ\n2) =\n\n\n\n\n\nsinh2 χ if k = −1,\n\nχ2 if k = 0 ,\n\nsin2 χ if k = −1,\n\n(2.11)\n\nhere χ plays a role as a radius or a new angular coordinate. It is sometimes useful\nto change the coordinate in time by defining the conformal time.\n\nη =\n\nˆ t dt′\n\na(t′)\n. (2.12)\n\nThen the FLRW metric becomes:\n\nds2 = a(η)2\n[\ndη2 − dχ2 − Sk(χ2)\n\n(\ndθ2 + sin2 θdϕ2\n\n)]\n. (2.13)\n\nWith this coordinate system, it is easy to see how the light propagates on null\ngeodesic ds2 = 0 with respect to the conformal time.\n\n2.4.2 Geodesic\n\nGeodesics are the shortest curve between two points. In such a curved space,\ntrajectories of free particles follow the geodesics with proper time τ such that:\n\nd2xµ\n\ndτ 2\n+ Γµαβ\n\ndxα\n\ndτ\n\ndxβ\n\ndτ\n= 0. (2.14)\n\nwhere xµ(τ) is the trajectories of free particles along the geodesics. The Christoffel\nsymbols Γµαβ are the metric connection.\n\nΓµαβ ≡\ngµν\n\n2\n[gαν,β + gβν,α − gαβ,ν ] =\n\ngµν\n\n2\n\n[\n∂gαν\n∂xβ\n\n+\n∂gβν\n∂xα\n\n− ∂gαβ\n∂xν\n\n]\n, (2.15)\n\nwhere (...) , µ ≡ ∂(...)\n\n∂xµ\n= ∂µ.\n\n\n\nGeneral relativity 20\n\n2.4.3 Einstein equations and Friedmann equations\n\nThe equivalent principle of classical mechanics stated the proportionality of the\ninertial and gravitational mass. It is equivalent to state that in a gravitational\nfield and in the absence of external forces, all masses fall at the same rate of\nacceleration. The Einstein tensor is described by the Ricci curvature tensor.\n\nGµν ≡ Rµν −\n1\n\n2\ngµνR. (2.16)\n\nWhere the Ricci curvature tensor 3 and Ricci scalar are given:\n\nRµν = Γαµν,α − Γαµα,ν + ΓαβαΓβµν − ΓαβνΓ\nβ\nµα,\n\nR ≡ gµνRµν , (2.17)\n\nThe commas means derivatives, for example Γαµν,α = ∂Γαµν/∂x\nα. The Einstein field\n\nequations link the curvature space-time (geometry) and the contents in matter\nand energy through the stress-energy tensor Tµν (illustrated in figure 2.10).\n\nRµν −\n1\n\n2\ngµνR− Λgµν = 8πGTµν . (2.18)\n\nG is the Newton constant. Λ is the cosmological constant that can be added to the\nequations. The stress-energy tensor describes the density and the flux of energy\nand momentum in space-time. The stress-energy tensor of the Universe in case of\na perfect fluid, in thermodynamic equilibrium is:\n\nT µν = Tµν =\n(\nρc2 + P\n\n) uµuν\nc2\n− Pgµν . (2.19)\n\nWhere we have introduced the 4-vector velocity uµ of the material object follow-\ning a time-line curve with respect to the proper time τ of the observer, so that\ngµνu\n\nµuν = −1. Then\n\nuµ =\ndxµ\n\ndτ\n. (2.20)\n\nHere ρ is the mass density and P is the pressure. Let us assume that the fluid is\nat rest uµ = (1, 0, 0, 0), uν = (−1, 0, 0, 0) then the stress-energy tensor of a perfect\n3The Ricci curvature tensor is the mathematical object that controls the growth rate of the\nvolume of metric balls in a manifold. In general relativity, the Ricci tensor is the part of the\nspace-time curvature which determines the volume of convergent or divergent matter in time.\nIt relates to the Riemann tensor Rα\n\nµαν ≡ Rµν .\n\n\n\nGeneral relativity 21\n\nfluid is:\n\nT µν =\n\n\n\n\nρc2 0 0 0\n\n0 P 0 0\n\n0 0 P 0\n\n0 0 0 P\n\n\n\n. (2.21)\n\nFigure 2.10: The intuitive components of the stress-energy tensor Tµν describe\nthe density, pressure, and momentum in spacetime.\n\nAssuming the Universe is homogeneous and isotropic we obtain the FLRW metric\n2.4.1 with coordinates xµ = (x0, x1, x2, x3) = (ct, r, θ, φ). The metrics FLRW in a\nspherical coordinate are:\n\ngµν =\n\n\n\n\n1 0 0 0\n\n0 − a2\n\n1− kr2\n0 0\n\n0 0 −a2r2 0\n\n0 0 0 −a2r2 sin2 θ\n\n\n\n. (2.22)\n\ngµν =\n\n\n\n\n1 0 0 0\n\n0 −1− kr2\n\na2\n0 0\n\n0 0 − 1\n\na2r2\n0\n\n0 0 0 − 1\n\na2r2 sin2 θ\n\n\n\n. (2.23)\n\nAfter solving the Einstein’s equations by calculating Christoffel, Ricci tensor (Appendix\n\nA). There are two independent Einstein equations, the time-time equation gives\nthe first Friedmann Equation relating the expansion of the Universe to the\n\n\n\nGeneral relativity 22\n\ndensity, the curvature and the cosmological constant.\n\nH2 ≡\n(\nȧ\n\na\n\n)2\n\n=\n8πG\n\n3c2\nρ− kc2\n\na2\n+\n\nΛc2\n\n3\n, (2.24)\n\nand\n\n2\nä\n\na\n+\n\n(\nȧ\n\na\n\n)2\n\n= −8πG\n\nc2\nP − kc2\n\na2\n. (2.25)\n\nBy subtracting the equation 2.25 to the first Friedmann equation, we get the\nsecond Friedmann equation, so-called acceleration equation\n\nä\n\na\n= −4πG\n\n3c2\n(ρ+ 3P ) +\n\nΛc2\n\n3\n. (2.26)\n\nThis implies that the acceleration of the expansion of the Universe is slowed down\nby the first term, gravity and speeded up by the second term, the cosmological\nconstant.\n\nThe contents of the Universe\n\nThese Friedmann equations describe the dynamics of the expanding Universe. We\nneed an equation describing the density and pressure of matter in the Universe.\nUnder the hypothesis that the expansion of the Universe is an adiabatic process\n(TdS ≡ δQ = 0) we have:\n\ndE + PdV = TdS. (2.27)\n\nThis is the first law of thermadynamics in volume V which has scale factor radius\na, density ρ. The energy in a co-moving volume is given by:\n\nE =\n4π\n\n3\na3ρc2. (2.28)\n\nIf the volume and density change with time then:\n\ndE\n\ndt\n= 4πa2ρ\n\nda\n\ndt\nc2 +\n\n4π\n\n3\na3dρ\n\ndt\nc2;\n\ndV\n\ndt\n= 4πa2da\n\ndt\n. (2.29)\n\nSubstitute into the equation 2.27, finally we can derive the fluid equation:\n\nρ̇+ 3\nȧ\n\na\n\n(\nρ+\n\nP\n\nc2\n\n)\n= 0 . (2.30)\n\n\n\nGeneral relativity 23\n\nThe equation of state is defined as\n\nP = ωρc2 ⇔ ω =\nP\n\nρc2\n. (2.31)\n\nThen the fluid equation become:\n\nρ ∝ a−3(1+ω) (2.32)\n\nCombine those equations above and the Friedmann equation [71], we have the\ntime evolution of the scale factor of the flat Universe.\n\na(t) ∝\n\n\n\n\nt2/3(1+ω) ω 6= −1,\n\neHt ω = −1,\n(2.33)\n\nIn case of:\nMatter dominated Universe ω = 0: a(t) ∝ t2/3, ρm ∝ a−3\n\nRadiation dominated ω =\n1\n\n3\n: a(t) ∝ t1/2, ρr ∝ a−4,\n\nCosmological constant dominated ω = −1: a(t) ∝ eHt, ρΛ ∝ a0,\n\nTable 2.1: The solutions of fluid equation and Friedmann equation for the\nUniverse in case of matter domination, radiation domination, cosmological dom-\n\nination.\n\nω ρ a\nMatter Domination 0 a−3 t2/3\n\nRadiation Domination\n1\n\n3\na−4 t1/2\n\nCosmological constant Λ -1 a0 eHt\n\nThe fate of the Universe\n\nThe Friedmann equation connects the density, the space curvature, and the ex-\npansion Hubble parameter. The critical density is defined as:\n\nρc ≡\n3c2H2\n\n8πG\n. (2.34)\n\nThen the density parameter as\nΩ =\n\nρ\n\nρc\n. (2.35)\n\n\n\nGeneral relativity 24\n\nThen the Friedmann equation becomes:\n\nΩ− 1 = − kc2\n\nH2a2\n+\n\nΛc2\n\n3H2\n. (2.36)\n\nThe density parameter for the cosmological constant:\n\nΩΛ =\nΛc2\n\n3H2\n. (2.37)\n\nThe density parameter for the curvature:\n\nΩk = − kc2\n\na2H2\n. (2.38)\n\nThe Friedmann equation now can be rewritten as:\n\nΩ + ΩΛ + Ωk = 1, (2.39)\n\nif the Universe contains mixing of matter, radiation and cosmological constant\ntoday. The density parameters are Ω = Ωm + Ωr + ΩΛ.\n\nFigure 2.11: Left: The geometry of the Universe depending on the total den-\nsity. Right: The fate of the Universe with respect to the components of dark\n\nmatter Ωm and dark energy ΩΛ.\nCredit: NASA/WMAP team\n\nThe value of Ω = 1 implies a flat universe, Ω < 1 an open universe, Ω > 1 a closed\nuniverse. The observational data imply that the density of the Universe today is\ncompatible with the critical density Ω0. Friedmann equations can be solved for\ndifferent Universe models as: Empty universe (Milne) Ω = 0; Einstein de\n\n\n\nGeneral relativity 25\n\nSitter Ω0 = Ωm;0 = 1; Super critical Ωm;0 = Ω0 > 1; Subcritical k = -1,\nΩm;0 = Ω0 < 1; Vacuum-dominated universes (de Sitter space) Only Λ,\nΩ0 = ΩΛ;0 = 1 this solution is useful to describe the inflationary Universe; Fluid\nk = 0, P = ρω; Flat k = 0, Ωm;0 + ΩΛ;0 = Ω0 = 1 . . . The evolution of scale factor\na(t) and time t depends on Ωm and ΩΛ is shown in figure 2.12\n\nFigure 2.12: The evolution of the scale factor a(t) and time t for several\nUniverse models, the blue dot is the de Sitter space, which is an exponential\nfunction of the Hubble parameter a(t) = eHt. The de Sitter space is a helpful\n\ncomputation describing the evolution during the inflation.\n\nThe Friedmann equation and the acceleration equation tell us that the evolution\nof the Universe depends on the expansion rate and the gravitational force. The\nHubble parameter measures the expansion rate of the Universe, while the gravity\nforce is determined by the density and pressure of the matter in the Universe.\nThe density parameter of radiation today is very small Ωr ≈ 10−4 and is obtained\n\n\n\nDistance 26\n\nby accurately measuring the temperature of the cosmic microwave background.\nThe latest measurements from the Planck mission indicate that h ∼ 0.678, Ωm ∼\n0.3,ΩΛ ∼ 0.7. Figure 2.11 illustrates the fate of the Universe in time and the scale\nfactor a(t).\n\nThe age of the flat Universe with Λ = 0\n\nAt current time t0, a(t0) = 1, H(t0) = H0. The equation 2.36 corresponds to:\n\nkc2 = H0(1− Ω0) (2.40)\n\nSubstituting the equation 2.40 into the Friedmann equation 2.24 we get:\n\nH2 =\n\n(\nda\n\nadt\n\n)2\n\n= H2\n0\n\n[\nΩra\n\n−4 + Ωma\n−3 + ΩΛ + (1− Ω0)a−2\n\n]\n. (2.41)\n\nUsing dt =\nda\n\naH\n, finally the age of the Universe is given by:\n\nt0 =\n\nˆ t0\n\n0\n\ndt =\n1\n\nH0\n\nˆ a(t0)=1\n\n0\n\nda√\nΩma−1 + Ωra−2 + ΩΛa2 + (1− Ω0)\n\n. (2.42)\n\nIn general, the integral should be computed numerically. In the special case of\nmatter dominated, the open Universe without vacuum energy ΩΛ = 0, Ωr ≈ 0 [71]\nis:\n\nt0 =\n2\n\n3\nH−1\n\n0 = 6.51h−1 × 109 years. (2.43)\n\nIn order to obtain the relation between the Hubble parameter and redshift, using\nthe definition in the equation 2.5 with a0 = 1 today then a = 1/(1+z), substituting\ninto the equation 2.41, we get the relationship [5]:\n\nH(z) = H0\n\n[\nΩm;0(1 + z)3 + Ωr;0(1 + z)4 + Ωk;0(1 + z)2 + ΩΛ;0\n\n]1/2\n. (2.44)\n\nFrom this equation 2.44 we can derive how the comoving distance relate to redshift,\nHubble constant and density parameters.\n\n\n\nDistance 27\n\n2.5 Distances\n\nIn cosmology, the measurement of distance in the expanding Universe is a very\nimportant point. Observational distances depend on the cosmological model of our\nUniverse. There have many ways to determine the distance between two points.\n\nComoving distance dχ\nThe fundamental distance is the distance between two points on a comoving grid.\nComoving distance is the distance from a distant emitter at redshift z, or scale\nfactor a and the observer at redshift z0 = 0, or scale factor a0 = 1 following\nformula:\n\nχ(a) =\n\nˆ t0\n\nt(a)\n\nc dt′\n\na(t′)\n;\n\n(2.45)\n\nWe can apply the changing variable:\n\ndt = da\ndt\n\nda\n=\n\nda\n\naH\n.\n\nThen we have:\nχ(a) =\n\nˆ 1\n\na\n\nc da′\n\na′2H(a′)\n. (2.46)\n\nFrom the equation 2.5 we have a = a0/(1+z), it is equivalent to: da = − a0 dz\n\n(1 + z)2\n,\n\nthen the cosmoving distance between us and an object at redshift z can be written\nas\n\nχ(z) =\n\nˆ z\n\n0\n\nc dz\n\nH(z)\n. (2.47)\n\nHere H(z) depends on matter contents of the Universe as shown in the equation\n2.44.\n\nThe angular diameter distance dA\nIn astronomy, if an object has diameter D and subtended angle θ, the angular\ndiameter distance is measured by:\n\ndA =\nD\n\nθ\n. (2.48)\n\nWe note that the comoving size of the object isD/a. So that the subtended angular\n\nsize related to comoving distance is θ =\nD/a\n\nχ(a)\n, substituting into the equation 2.48,\n\n\n\nThe horizon problem 28\n\nwe show that the angular diameter distance is\n\ndA = a χ(a) =\na0χ(a)\n\n1 + z\n. (2.49)\n\nThe luminosity distance dL\nThe luminosity distance is defined as the flux of an object known as luminosity L\n(The energy emitted in a solid angle 4π) and flux density S.\n\n(dL)2 =\nL\n\n4πS\n. (2.50)\n\nLet us consider an object at scale factor a, r0 ≡\n´ t0\nt(a)\n\nc dt/a(t) is the comoving\nradius, then the radiation flux accounting for the expansion of the Universe S =\n\nL/(4πa2r2\n0(1 + z)2) [71]. Therefore the equation 2.50 becomes:\n\ndL = ar0(1 + z). (2.51)\n\nFor nearby objects z � 1, the luminosity distance is the physical distance.\n\n2.6 The horizon problem\n\nThe horizon problem can be understood as the homogeneity problem, the commu-\nnication between two opposite regions in the observable Universe. The comoving\nparticle horizon (equivalent to the conformal time) η or the maximum distance a\nphoton can travel between 0 and time t in the Universe is [7]\n\nη =\n\nˆ t\n\n0\n\ndt′\n\na(t′)\n=\n\nˆ a\n\n0\n\nda′\n\na′\n1\n\na′H(a′)\n=\n\nˆ a\n\n0\n\nd ln a′\n1\n\na′H(a′)\n∝\n\n\n\n\na radiation dominated.\n\na1/2 matter dominated.\n(2.52)\n\nThe comoving horizon is the logarithmic integral of the comoving Hubble radius\n1/aH. If we consider two points on the last scattering (LS) surface, as illustrated\nin figure 2.13, corresponding to looking at the opposite direction in the sky, these\nregions would have not time to interact on the opposite side of the sky. The\nobservational CMB temperature is nearly isotropic at 2.725 K with tiny fluctua-\ntions 10−5. In the matter-dominated epoch, the horizon distance of two regions is\n\n\n\nThe flatness problem 29\n\napproximately [129].\n\ndhor(tLS) = 2\nc\n\nH(tLS)\n∼ 0.4Mpc. (2.53)\n\nThe angular diameter distance to the last scattering surface is dA ≈ 14Mpc [129]\nthen the angular between two points is:\n\nθhor =\ndhor(tLS)\n\ndA\n∼ 2◦. (2.54)\n\nt0\n\ntrec\nti\n\nOBSERVER\n\nA BRECOMBINATION\n\nSINGULARITY\n\nCMB PHOTON\n\nFigure 2.13: The horizon problem, two points A and B at the recombination\nepoch, there is not time for the signal to travel between A and B. How they can\n\nhave the same temperature with tiny fluctuations?\n\nIt means that points separated by more than 2◦ on the last scattering surface\ndo not have time to interact with each other. However, the angular scale on the\nCMB map is separated by about 1◦. So why does the universe look the same in\nall directions? The inflationary scenario which is proposed by Alain Guth solves\nthis problem. The idea of inflation scenario is that actually, two regions started\nreally close to each other in the very early Universe, then cosmological inflation\nhas driven the Universe expanded exponentially quickly. The detail ideal and the\nsolution for the horizon problem are described in section 2.8.\n\n\n\nInflation 30\n\n2.7 The flatness problem\n\nIn the flat Universe (k=0). From the Friedmann equation, the critical density is\n\nρc(t) =\n3H2\n\n8πG\n(t) = 1.88h2 × 10−26 kgm−3. (2.55)\n\nWe know that the total density of material in the Universe is closely related to\nthe critical density. The Friedmann equation 2.24 can be rearranged as\n\n1− Ωtot(t) =\n−k\n\n(aH)2\n. (2.56)\n\nwhere the density parameter is already defined as Ωtot(t) ≡\nρ(t)\n\nρc(t)\n. The case\n\nΩtot(t) = 1 is an unstable point. Indeed, if the Universe is flat Ωtot = 1, then\nit remains for all time and independent to the comoving Hubble radius (aH)−1\n\nwhich grows with time. The observation of type Ia supernovae, BAO and the\nCMB show that 1 − Ωtot(a) ∼ 0.005 today. Why Ωtot is so close to 1? Why is\nit not smaller or lager? Again, the rapid exponential expansion in the inflation\nscenario solves the flatness problem.\n\n2.8 Inflation\n\nIn 1981, Alan H. Guth published a paper \"A possible solution to the horizon and\nflatness problems\" [47]. The author proposed a scenario of the inflating Universe\nwith a huge expansion factor, and the scale factor is an exponential function of\ntime. This solution implies a negative pressure of the Universe. Particles of whole\nobservational Universe were causally connected together. From the acceleration\nequation 2.26 we have the requirement of inflation:\n\nd\n\ndt\n\n1\n\naH\n< 0,⇒ d2a\n\ndt2\n> 0,⇒ P < −ρc\n\n2\n\n3\n. (2.57)\n\nDuring the inflation, the early Universe behaves as it was dominated by a cosmo-\nlogical constant, then the Friedman equation gives:\n\nH2 =\nΛ\n\n3\n, (2.58)\n\n\n\nInflation 31\n\nThe Hubble parameter is then constant over the time, then the acceleration equa-\ntion is\n\nä\n\na\n=\n\nΛ\n\n3\n, (2.59)\n\nand the solution for the scale factor is\n\na(t) = exp\n\n(√\nΛ\n\n3\nt\n\n)\n= exp(Ht). (2.60)\n\nFrom this equation 2.60 we can assume the exponential inflation starts at ti and\nends at tf , the scale factor is\n\na(tf )\n\na(ti)\n= eN . (2.61)\n\nWhere N is the number of e-foldings of inflation:\n\nN ≡ H(tf − ti) (2.62)\n\nFor N=60 at least required to explain observational facts, then\n\na(tf )\n\na(ti)\n∼ e60 ' 1025. (2.63)\n\nThis is a huge expansion factor of the early Universe. This also provides expla-\nnation of the horizon problem today. Two opposite regions were actually causally\nconnected at the inflation epoch.\n\nSolution of the flat problem\n\nFrom the equation 2.36 and 2.56, if the Universe is exponentially expanding during\nthe inflation, then [129]\n\n1− Ω(t) ∝ e−2Ht. (2.64)\n\nNow we consider the beginning and ending of the inflation, from equation 2.62 we\nhave t = tf = ti +N/H, substituting into equation 2.64 we obtain:\n\n1− Ω(tf ) = e−2N(1− Ω(ti)). (2.65)\n\nAfter 60 e-foldings e−2N ' 10−52, even if at the beginning of the inflation the\nUniverse is not flat, the inflation flatten the Universe by factor of ∼ 1052. The\ntype Ia supernovae and the CMB observation data indicate that the current limits\non the density parameter is 1 − Ω ≤ 0.005, it implies that the inflation happens\nwith e-fold > 60.\n\n\n\nPhysics of inflation 32\n\nSolution of the horizon problem\n\nAt any time t′, the horizon distance is with respect to the comoving particle\nhorizon:\n\ndhor(t) = a(t)c\n\nˆ t\n\n0\n\ndt′\n\na(t′)\n. (2.66)\n\nAssuming that the beginning of the inflation is radiation-dominated a(ti) ∝ t\n1/2\ni ,\n\nthen the horizon distance at the beginning of the inflation is:\n\ndhor(ti) = a(ti)c\n\nˆ ti\n\n0\n\ndt\n\na(t)\n= ct\n\n1/2\ni\n\nˆ ti\n\n0\n\ndt\n\nt1/2\n= 2cti, (2.67)\n\nthen the horizon distance at the end of the inflation is calculated as, following\nequations 2.60 and 2.61:\n\ndhor(tf ) = a(tf )c\n\nˆ tf\n\n0\n\ndt\n\na(t)\n\n= a(ti)e\nNc\n\n[ˆ ti\n\n0\n\ndt\n\nt1/2\n+\n\nˆ tf\n\nti\n\ndt\n\na(t)\n\n]\n;\n\n= eNct\n1/2\ni\n\n[\n2t\n\n1/2\ni +\n\nˆ tf\n\nti\n\ndt\n\neH(t−ti)\n\n]\n;\n\n= eNc\n\n(\n2ti +\n\nt\n1/2\ni\n\nH\n\n(\n1− e−N\n\n)\n)\n\n(2.68)\n\nThe horizon is boosted by an exponentional factor. For one possible model, the in-\nflation that it statred around the Grand Unified Theory (GUT) time, ti ≈ tGUT ≈\n10−36s, with the Hubble parameter Hi ≈ t−1\n\nGUT . The horizon dhor(ti) = 2cti ≈\n6× 10−28m [129], the Hubble parameter is H ≈ 1036s−1 for e-folding ≈ 100. Then\n\nthe horizon after the inflation is immediately dhor(tf ) ≈ eNc2ti\n\n(\n1 +\n\nt\n1/2\ni\n\n2\n\n)\n≈\n\n8× 1015m ≈ 1.5pc.\n\n2.9 Physics of inflation\n\nIn many inflation models, the simplest way to explain the mechanism of inflation is\nproduced by a scalar field φ, the inflaton, which is a function of position and time.\nThe dynamic of a scalar field and gravity is the gravitational Einstein-Hilbert\naction4 (in a physical system the equation of motion is derived by a dynamic\n4The action was first proposed by Hilbert, it attributes to the Einstein field equations\n\n\n\nPhysics of inflation 33\n\nFigure 2.14: An example of a scalar field of inflation. The inflation happens\nwhen the potential energy accelerates the field. The quantum fluctuations im-\nprint fluctuations on the CMB. The density energy of inflation is transferred\n\ninto radiation at reheating oscillation around the minimum [18].\n\nattribution) of general relativity and scalar field action [18]. Let us take the\nEinstein-Hilbert action and a scalar field:\n\nS =\n\nˆ\nd4x\n√−g\n\n[\nM2\n\nPl\n\n2\nR− 1\n\n2\ngµν∂µφ∂νφ− V (φ)\n\n]\n= SEH + Sφ. (2.69)\n\nwhere g = det(gµν), R = gµνRµν is the Ricci scalar, V (φ) is the potential energy\n\nof the scalar field, the reduced Planck mass MPl ≡\n1√\n8π\nmPl ≡\n\n√\n~c√\n\n8πG\n= 2.436×\n\n1018GeV (the Planck mass mPl =\n\n√\n~c\nG\n, ~ is the reduced Planck constant) and\n\nalso we can see the starting point in the field theory, the function of space-time,\nLagrangian density which is an action of a canonical kinetic term and a scalar\nfield, Sφ:\n\nL = −1\n\n2\ngµν∂µφ∂νφ− V (φ). (2.70)\n\nThe stress-energy tensor of the evolution of space-time for the scalar field is [18]\n\nT (φ)\nµν ≡ −\n\n2√−g\nδSφ\nδgµν\n\n= ∂µφ∂νφ− gµν\n(\n\n1\n\n2\n∂σφ∂σφ+ V (φ)\n\n)\n. (2.71)\n\nIf the inflation field is homogeneous (the gradient of the inflation field equals 0,\n∇φ = 0,∇2φ = 0) during the inflation then the energy density and the pressure\nare:\n\nρφ = T00 =\n1\n\n2\nφ̇2 + V (φ),\n\n\n\nPhysics of inflation 34\n\nPφ =\n1\n\n3\n(T11 + T22 + T33) =\n\n1\n\n2\nφ̇2 − V (φ). (2.72)\n\nWe also can get the equation of state\n\nωφ =\nPφ\nρφ\n\n=\nφ̇2 − 2V (φ)\n\nφ̇2 + 2V (φ)\n. (2.73)\n\nThe equation of state shows that if the potential energy dominates the kinetic of\na scalar field, negative pressure is possible. We can obtain the useful combination:\n\nρφ + Pφ = φ̇2,\n\nρφ + 3Pφ = 2(φ̇2 − V (φ)). (2.74)\n\nThe equation of motion for the scalar field, assuming the FLRW metric, flat space\nand homogeneity, is [137]\n\nδS\n\nδφ\n\n1√−g∂µ(\n√−g∂µφ) + V̇ (φ) = 0.\n\n⇒ φ̈+ 3Hφ̇+ V̇ (φ) = 0 . (2.75)\n\nThe term 3Hφ̇ plays a role of friction, it slows down the motion as well as the\nevolution of the inflaton field in the equation of motion. It is an attractor \"slow-\nroll\".\n\nIn order to describe the energy of the scalar field during the inflation, inserting in\nthe equation 2.72 into 2.24 we have.\n\nH2 =\n1\n\n3M2\nPl\n\n(\n1\n\n2\nφ̇2 + V (φ)\n\n)\n. (2.76)\n\nThe equation 2.75 and 2.76 are called the equations of motion and the scalar field\nequation.\n\n2.9.1 Slow-Roll inflation\n\nThe standard approximation to obtain slow-roll parameters is to neglect the small\nterms in the equation 2.75 and 2.76. They are considered as conditional equations:\n\n3Hφ̇ ' −V̇ (φ), (φ̈� 3Hφ̇)\n\n\n\nPhysics of inflation 35\n\nH2 ' V (φ)\n\n3M2\nPl\n\n. (φ̇2 � V (φ)) (2.77)\n\nThe slow-roll parameters have been introduced by Liddle and Lyth [72].\n\nε(φ) =\nM2\n\nPl\n\n2\n\n(\nV̇\n\nV\n\n)2\n\n� 1.\n\n|η(φ)| =\n∣∣∣M2\n\nPl\n\nV̈\n\nV\n\n∣∣∣� 1. (2.78)\n\nThese conditions constrain the shape of the potential energy of the scalar field.\nThe second parameter implies an attractor solution, which starts from arbitrary\ninitial conditions to the basin of attraction by the attractor as illustrated in figure\n2.14. These parameters also guarantee a long lived enough inflation for more than\n60 e-folds. The first parameter implies a background solution related to the Hubble\nrate, we can see that by revisiting the condition in the accelerating equation of\nthe inflation.\n\nH =\nȧ\n\na\n⇒ Ḣ =\n\nä\n\na\n− ȧ2\n\na2\n⇒ ä\n\na\n= Ḣ +H2 > 0⇒ − Ḣ\n\nH2\n< 1. (2.79)\n\nThe slow-roll parameter relates to the Hubble parameter, from the conditional\nequations 2.77 and remembering that 3Hφ̇ = −V̇ , we have:\n\nH2 =\nV\n\n3M2\nPl\n\nderivation⇒ 2HḢ =\nV̇ φ̇\n\n3M2\nPl\n\n⇔ H2Ḣ =\nV̇ Hφ̇\n\n6M2\nPl\n\n= − V̇ 2\n\n18M2\nPl\n\n,\n\n⇒ − Ḣ\n\nH2\n=\n\n1\n\n2\nM2\n\nPl\n\n(\nV̇\n\nV\n\n)2\n\n= ε� 1. (2.80)\n\nIt is useful to notice that:\n\nH =\nȧ\n\na\n⇒ ȧ = aH,\n\nä = ȧH + aḢ = a(H2 + Ḣ),\n\n= aH2(1− ε) (2.81)\n\nSo during slow-roll inflation ä ∼ aH2 and inflation ends when ε = 1. The Hubble\nparameter slowly changes comparing to the scalar factor a. Number of e-folds\nfrom time ti until to the inflation finish tf is:\n\nN ≡ ln\na(tf )\n\na(ti)\n,\n\n\n\nPrimordial quantum fluctuations in inflation and cosmological perturbations 36\n\n=\n\nˆ tf\n\nti\n\nH(t)dt =\n\nˆ φf\n\nφi\n\nH\n\nφ̇\ndφ =\n\nˆ φf\n\nφi\n\n3H2\n\n−V̇\ndφ,\n\n=\n1\n\nM2\nPl\n\nˆ φi\n\nφf\n\nV\n\nV̇\ndφ. (2.82)\n\nwhere dt =\ndφ\n\nφ̇\n, φ̇ =\n\n−V̇\n3H\n\n. The fluctuations in the CMB provide constrain on value\n\nof number e-folds between NCMB ≈ 40− 60 for some specific models. The precise\nvalue depends on scenarios of the inflation as shown in figure 2.15.\n\n0.94 0.96 0.98 1.00\n\nPrimordial tilt (ns 0.002)\n\n0.\n00\n\n0.\n0\n5\n\n0.\n10\n\n0\n.1\n\n5\n0\n.2\n\n0\n\nT\nen\n\nso\nr-\n\nto\n-s\n\nca\nla\n\nr\nra\n\nti\no\n\n(r\n0\n.0\n\n0\n2\n)\n\nConvex\n\nConcave\n\nTT,TE,EE+lowE+lensing\n\nTT,TE,EE+lowE+lensing+BK14\n\nTT,TE,EE+lowE+lensing+BK14+BAO\n\nNatural inflation\n\nHilltop quartic model\n\nα attractors\n\nPower-law inflation\n\nR2 inflation\n\nV ∝ φ2\n\nV ∝ φ4/3\n\nV ∝ φ\nV ∝ φ2/3\n\nLow scale SB SUSY\nN∗=50\n\nN∗=60\n\nFigure 2.15: The theoritical inflation scenarios (Planck 2018 result) are con-\nstrained by Planck data and BAO, BICEP [109, 118].\n\n2.10 Primordial quantum fluctuations in inflation\n\nand cosmological perturbations\n\nHomogeneity and isotropy of the Universe on large-scale is an usual assumption.\nOn a microscopic scale as the human, the Earth, the Solar system, the stars even\ngalaxy clusters, the Universe is highly inhomogeneous [66]. Recent observations of\nthe Planck satellite measured CMB anisotropics on the full sky with unprecedented\naccuracy. It observed tiny fluctuations of the temperature ∆T/T ∼ 10−5 as shown\nin figure 2.16. It means that at the epoch of recombination, the Universe was nearly\nperfectly homogeneous. These fluctuations of the early Universe are the seed\nof the large-scale structure and originate from cosmological perturbations. The\n\n\n\nPrimordial quantum fluctuations in inflation and cosmological perturbations 37\n\nFigure 2.16: The CMB anisotropies implies the inhomogeneous Universe. The\nPlank CMB maps measured a tiny fluctuation in order of 10−5. The structure\nformation of the Universe today can be analyzed by a linear perturbation of\n\ninflation field. Credit: ESA/Planck team.\n\nquestion is what created those tiny primordial fluctuations observed on CMB? The\ncomplex structure formation of the Universe is the consequence of the generation\nand evolution of inhomogeneities. Therefore there are two parts in the theory:\n\n1. The generation of inhomogeneity is speculated by primordial quantum per-\nturbations in the very early Universe. The initial quantum fluctuations of\nthe scalar field φ seed the large-scale structure today.\n\n2. The growth of the inhomogeneity is predicted by perturbations of the metric,\nby gravitational amplification and the effect of the pressure force in the\nframe of the general relativity depending on the equation of state, density\nparameters.\n\n2.10.1 Linear perturbation\n\nBecause the inhomogeneity of the Universe is small we can describe those with\nlinear perturbations around homogeneity. The linear perturbation is a beautiful\nand exciting way to treat the initial quantum fluctuations from the inflation field\ninto the macroscopic cosmological perturbations. A quantities X(t,x) of time\n\n\n\nPrimordial quantum fluctuations in inflation 38\n\nand coordinate such as metric gµν or stress-energy Tµν ( → φ, ρ, P, . . .) can be\ndecomposed as a homogeneous background X̄(t) plus a perturbation:\n\nδX(t,x) ≡ X(t,x)− X̄(t). (2.83)\n\nBecause a transformation of the time coordinate can introduce fictitious pertur-\nbations. The split into background and perturbations depends on the chosen\ncoordinates or gauge choice which is precise the transformation between differen-\ntial geometry [18]. The perturbation for metric, energy density and pressure can\nbe written:\n\ngµν(t,x) = ¯gµν(t) + δgµν(t,x).\n\nρ(t,x) = ρ̄(t) + δρ(t,x).\n\nP (t,x) = P̄ (t) + δP (t,x). (2.84)\n\nIn order to simplify the differential equations of perturbations, the computation\nis performed in Fourier space with independent Fourier modes (different wave\nnumber) which can be studied independently:\n\nXk(t) =\n\nˆ\nd3xX(t,x)eik·x. (2.85)\n\nHere X ≡ δφ, δgµν , δρ, δP and notice that k is wave numbers and k is wave vectors.\n\n2.10.2 Primordial quantum fluctuations in inflation\n\nLet us discuss briefly the generation of primordial quantum perturbations in in-\nflation. The detail calculation of generation and evolution of the perturbation are\navailable in [16–18, 66, 68, 72, 75, 137]. The generation of primordial perturba-\ntions in the framework of inflation are due to quantum fluctuations of the motion\nof the scalar field, which is the source of tensor, scalar power spectra perturbations\nPt(k), Ps(k). The perturbation during inflation is defined as:\n\nφ(t,x) = φ̄(t) + δφ(t,x).\n\n(2.86)\n\nInserting the equation 2.86 into the motion equation 2.75 (generally we have to\nkeep the gradient term ∇2φ), we get the field equation for a scalar field in FRLW\n\n\n\nPrimordial quantum fluctuations in inflation 39\n\nspace with homogeneous background elements and perturbation elements:\n\n(\nφ̄+ δφ\n\n)..\n+ 3H\n\n(\nφ̄+ δφ\n\n). − a−2∇2\n(\nφ̄+ δφ\n\n)\n+ V̇\n\n(\nφ̄+ δφ\n\n)\n= 0. (2.87)\n\nScalar perturbations in inflation (de sitter space-only Λ): Harmonic\n\nOscillations\n\nAs mention before, if the inflation field is homogeneous during the inflation, the\ngradient of the inflation field equals 0, thus ∇φ = 0. By subtracting the equation\n2.87 to the background equation 2.75 with notice that V̇ (φ̄ + δφ) = V̇\n\n(\nφ̄\n)\n\n+\n\nV̈\n(\nφ̄\n)\nδφ, we have the perturbation equation during inflation:\n\nδφ̈+ 3Hδφ̇− a−2∇2δφ+ V̈\n(\nφ̄\n)\nδφ = 0, (2.88)\n\nin Fourier domain:\n\nH−2δφ̈k + 3H−1δφ̇k +\n\n[(\nk\n\naH\n\n)2\n\n+\nm2\n\nH2\n\n]\nδφk = 0. (2.89)\n\nWhere m2(φ) ≡ V̈ (φ̄). During inflation, H and m2 change slowly, then we can\nignore them2/H2 in the slow-roll approximation (m2 � H2). The general solution\nof the second order different equation of perturbations is:\n\nδφk(t) = Akωk(t) +Bkω\n∗\nk(t), (2.90)\n\nwith\nωk(t) =\n\n(\ni+\n\nk\n\naH\n\n)\nexp\n\n(\nik\n\naH\n\n)\n. (2.91)\n\nThe solution implies that before the Hubble horizon exit (k � aH ), as a(t)\n\nincrease the oscillations are rapid and after the Hubble horizon exit (k � aH) the\noscillations approach to constant values i (Ak +Bk).\n\nIn fact, those calculations above ignored the metric perturbations, however, per-\nturbations depend on the choice of coordinates called gauge choice. We introduce\ngauge transformation for the curvature perturbation R which defines the gauge\ninvariant curvature perturbation [18]\n\nR = −Hδφ\n˙̄φ\n. (2.92)\n\n\n\nPrimordial quantum fluctuations in inflation 40\n\nUsing statistical properties of Gaussian perturbation (which has a stable expec-\ntation and a variance) and computational quantum mechanics of the harmonic\noscillator [18, 75], the power spectrum describes completely statistical properties\nof random perturbations of the perturbed universe. During inflation, the ampli-\ntude of fluctuation scales with Hubble parameter H. Then we choose the value of\naH = k at the horizon exit. Finally, the power spectrum of a scalar perturbation\nis given by:\n\nPR(k) =\n\n(\nH\n˙̄φ\n\n)2\n\nPφ(k) =\n\n(\nH\n˙̄φ\n\n)2(\nV\nk3\n\n2π2\n〈|δφk|2〉\n\n)\n=\n\n(\nH\n˙̄φ\n\n)2(\nH\n\n2π\n\n)2\n\n. (2.93)\n\nThis scalar primordial spectrum is assumed for the calculation of structure forma-\ntion and the CMB anisotropy.\n\nTensor perturbations in inflation (de sitter space-only Λ): Primordial\n\ngravitational waves\n\nSimilarly, during inflation, we start from the Einstein-Hilbert gravity action plus\nthe matter action. We can rewrite equation 2.69 as:\n\nS ≡\nˆ\nd4x\n\n\n Lg︸︷︷︸\ngravity\n\n+ Lm︸︷︷︸\nmatter\n\n\n =\n\nM2\nPl\n\n2\n\nˆ\nd4x\n√−gR\n\n︸ ︷︷ ︸\nEinstein−Hilbert action\n\n+\n\nˆ\nd4x\n√−g\n\n[\n−1\n\n2\ngµν∂µφ∂νφ− V (φ)\n\n]\n\n︸ ︷︷ ︸\nMatter action\n\n.\n\n(2.94)\nThe metric tensor including a small tensor perturbation hµν is:\n\ngµν = ḡµν + hµν . (2.95)\n\nFrom the FLRW metric we can defined 3×3 tensor hij as transverse and traceless.\n\nds2 = a2(η)\n[\n−dη2 + (δij + hij) dx\n\nidxj\n]\n. (2.96)\n\nHere η is the conformal time, a is the scale factor. We obtain the second-order\naction for tensor perturbations due to the fact that the first order is gauge-invariant\n[18, 151]. From the second-order action, we can calculate the stress-energy tensor\nand the equation of motion.\n\n(2)S =\nM2\n\nPl\n\n8\n\nˆ\ndηd3xa2\n\n[(\nḣij\n\n)2\n\n− (∇hij)2\n\n]\n. (2.97)\n\n\n\nPrimordial quantum fluctuations in inflation 41\n\nThe linearized perturbation is applied in Einstein equations [90]:\n\nδ\n\n[\nRµν −\n\n1\n\n2\nRgµν\n\n]\n= 8πGδTµν . (2.98)\n\nThe stress-energy tensor is calculated by integration of the distribution function\nof momentum space using the action 2.97. The transverse and traceless condition\nin a spatial metric perturbation is ∂ihij = hii = 0. Finally the solution of the\nequation 2.98 is the equation of motion [8, 90, 151]:\n\nḧij + 2Hḣij −∇2hij = 16πGa2Πij. (2.99)\n\nWhere H is the comoving Hubble parameter, H =\n1\n\na\n\nda\n\ndη\n= aH = k. Πij is\n\nthe transverse (∂iΠij = 0) and traceless (Πii = 0) components of the energy\nmomentum tensor.\n\na2Πij = Tij − pgij. (2.100)\n\nLikewise we did for a scalar perturbation, we introduce two polarization states h+,\nh× and we perform the Fourier expansion\n\nhij(x, η) =\n1\n\n(2π)3\n\nˆ\ndk eik·x\n\n[\nh+e\n\n+\nij + h×e\n\n×\nij\n\n]\n. (2.101)\n\nHere e+\nij, e\n\n×\nij denote for two symmetric polarization tensors, their properties are\n\n[90]:\n\nkie+\nij = kie×ij = 0, e+,i\n\ni = e×,ii = 0,\n\ne+\nije\n\n+,ij = e×ije\n×,ij = 2, e+\n\nije\n×,ij = 0. (2.102)\n\nIn the cartesian coordinate, polarization tensors are [48]:\n\ne1 =\n1√\n2\n\n\n\n\n1 0 0\n\n0 −1 0\n\n0 0 0\n\n\n and e2 =\n\n1√\n2\n\n\n\n\n0 1 0\n\n1 0 0\n\n0 0 0\n\n\n . (2.103)\n\nThe left- and right-handed polarizations can be basically defined as:\n\ne+,× ≡ e1 ± ie2. (2.104)\n\nThe simplest way to solve equation 2.99 is that if the inflationary expansion is\n\n\n\nPrimordial quantum fluctuations in inflation 42\n\ndriven by a scalar field, since then the energy-momentum tensor is absented. Ap-\nplying the Fourier transform of hij(x, η) and substituting again in the motion equa-\ntion 2.99 (the Laplace operator transforms in Fourier transformation as ∇ → −k2)\nwe have:\n\nḧk + 2Hḣk + k2hk = 0. (2.105)\n\nHere\nhk =\n\nH\n\nMPl\n\nie−iktk−3/2, (2.106)\n\nit is now clear that motion equation of tensor perturbations 2.105 solves the wave\nequation, hence the solution gives primordial gravitational waves. To solve this\nequation, it is useful to use the approximation of the de Sitter space. Finally the\nscale of the power spectrum of tensor perturbation of each polarization mode is\ngiven by [18, 26, 72, 90].\n\nPh(k) ≡ k3\n\nπ2\n\n(\n|h+|2 + |h×|2\n\n)\n=\n\n4\n\nM2\nPl\n\n(\nH\n\n2π\n\n)2\n\n(2.107)\n\nGravitational waves are tensor perturbations of the metric and these signal im-\nprint in the polarization of the CMB. Many CMB experiments aim to probe these\nprimordial gravitational wave through the B-mode signal.\n\nWe already established the scalar perturbation equation 2.93:\n\nPs(k) ≡ PR(k) =\nH2\n\n(2π2)\n\nH2\n\nφ̇2\n=\n\n1\n\n8π2\n\nH2\n\nM2\nPl\n\n1\n\nε\n\n∣∣∣∣∣\nk=aH\n\n. (2.108)\n\nThe power spectrum of tensor perturbations for two polarization modes is calcu-\nlated by equation 2.107:\n\nPt(k) = 2Ph(k) =\n2\n\nπ2\n\nH2\n\nM2\nPl\n\n∣∣∣∣∣\nk=aH\n\n. (2.109)\n\nThe tensor-to-scalar ratio is defined as:\n\nr =\nPt(k)\n\nPs(k)\n= 16ε. (2.110)\n\n\n\nCosmological perturbations and structure formation 43\n\nThe scale dependence parameters of the spectra are:\n- the scalar spectral index:\n\nns − 1 =\nd lnPs(k)\n\nd ln k\n. (2.111)\n\n- the tensor spectral index:\n\nnt =\nd lnPt(k)\n\nd ln k\n. (2.112)\n\nwhere ln k = N +lnH and ε = −d lnH\n\ndN\n, η = −\n\nd ln\ndH\n\ndφ\n\ndN\n. In the standard slow-roll\n\napproximation:\n\nns − 1 = 2η − 6ε,\n\nnt = −2ε,\n\nr = 16ε = −8nt (2.113)\n\nThe Lyth bound provides relationship directly between the tensor-to-scalar and\nthe number of e-folds [74]:\n\nr =\n8\n\nM2\nPl\n\n(\ndφ\n\ndN\n\n)2\n\n. (2.114)\n\nThe energy scale of inflation is directly linked to the tensor-to-scalar [18].\n\nV 1/4 ∼\n( r\n\n0.01\n\n)1/4\n\n1016GeV. (2.115)\n\nThe measurement of tensor-to-scalar is the main target of the modern cosmology.\nFrom its value, we can extract those parameters and understand the inflation\nmodel.\n\n2.10.3 Cosmological perturbations and structure formation\n\nSo far, we described the quantum fluctuations in inflation epoch in the very early\nUniverse, these fluctuations seed the growth of structures formation and the evolu-\ntion of large-scale structures. The linear perturbations approach is applied to treat\ncosmological perturbations. The relativistic perturbation theory is a fully general\nrelativistic treatment of cosmological perturbations in which we can treat pertur-\nbations of the metric, the comoving curvature, the scalar, the vector, the tensor\n\n\n\nCosmological perturbations and structure formation 44\n\nand the matter (the stress-energy tensor). Perturbations evolve in the primordial\nplasma from the inflation to the CMB emission.\n\nFigure 2.17: Evolution of density perturbation of photons, baryons and dark\nmatter [17].\n\nFigure 2.17 presents the evolution of baryons, photons and Cold Dark Matter\n(CDM) with respect to time in two different wave-numbers k. Before the de-\ncoupling epoch z > zdec ≈ 1100, the radiation era, baryons, and photons are\ncoupled strongly by Compton scattering as a single fluid, on small scales (the\nlarge wave-number k) the radiation pressure exceed by photons, pressure opposes\nthe squeezing or compression of the plasma fluid inducing oscillations are called\nsound waves. Baryon-photon fluid oscillates in the potential wells of dark matter,\nbut fluctuation amplitudes are small of the order of dT/T ∼ 1 part in 105. Dark\nmatter is not coupled to photons and baryons (except through gravity), so its fluc-\ntuations can grow independently. Just after the decoupling, the baryons fall into\nthe potential wells of the grown dark matter density. Radiation is free-streaming\nafter recombination. With the large scale (the smaller wave number) these fluctu-\nation amplitudes of oscillations is invisible due to pressure effects can be neglected.\n\n\n\nCMB 45\n\nDark matter perturbations evolve with different wavelengths. In general, the evo-\nlution of the gravitational potential Φ is sourced by the total density fluctuations\nin the radiation era, radiation-to-matter transition era, and matter era. The grav-\nitational potential is constant on all scales during matter domination [17]. In the\nradiation era, the perturbed radiation density is the acoustic oscillations which are\npeaks in the CMB temperature anisotropies spectrum.\n\n\n\nChapter 3\n\nThe Cosmic Microwave Background\n\n(CMB)\n\nContents\n3.1 The CMB . . . . . . . . . . . . . . . . . . . . . . . . . . . 49\n\n3.2 Physics of CMB temperature anisotropies . . . . . . . 60\n\n3.3 CMB polarization . . . . . . . . . . . . . . . . . . . . . . 63\n\n3.4 Primordial non-Gaussianity in the CMB . . . . . . . . 68\n\n3.5 Gravitational lensing . . . . . . . . . . . . . . . . . . . . 69\n\n3.6 CMB spectral distortions . . . . . . . . . . . . . . . . . . 70\n\n3.7 Foreground components . . . . . . . . . . . . . . . . . . . 72\n\n3.7.1 Thermal dust . . . . . . . . . . . . . . . . . . . . . . . . 74\n\n3.7.2 Synchrotron . . . . . . . . . . . . . . . . . . . . . . . . . 75\n\n3.7.3 Free-free . . . . . . . . . . . . . . . . . . . . . . . . . . . 76\n\n3.7.4 Spinning dust . . . . . . . . . . . . . . . . . . . . . . . . 76\n\n3.8 Systematic effects . . . . . . . . . . . . . . . . . . . . . . 76\n\n3.8.1 Cosmic rays . . . . . . . . . . . . . . . . . . . . . . . . . 78\n\n3.8.2 Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78\n\n3.8.3 Bandpass mismatch . . . . . . . . . . . . . . . . . . . . 80\n\n3.9 State of the art . . . . . . . . . . . . . . . . . . . . . . . . 80\n\n3.10 QUBIC and LiteBIRD . . . . . . . . . . . . . . . . . . . 83\n\n3.10.1 Ground base experiment: QUBIC . . . . . . . . . . . . . 83\n\n46\n\n\n\nThe Cosmic Microwave Background (CMB) 47\n\n3.10.1.1 General principle . . . . . . . . . . . . . . . . . 83\n\n3.10.1.2 Instrument . . . . . . . . . . . . . . . . . . . . 87\n\n3.10.2 Space satellite mission: LiteBIRD . . . . . . . . . . . . . 88\n\nThe evolution of the Universe is illustrated in figure 2.2 under the Big Bang theory.\nThe Cosmic Microwave Background (CMB) is radiation from around 380 000 years\nafter the Universe was born. Before this time, the Universe was so hot, dense and\nopaque, the early Universe was made of plasma of matter and radiation. Thus\nphotons could not travel freely and no light escaped from those earlier times. The\nCMB was emitted at the recombination era where electrons combined with p and\nHe nuclei atoms and photons were suddenly free to propagate in all directions.\n\ne+ p\n H + γ.\n\nDuring a small interval time, the Universe suddenly switched from opaque to com-\npletely transparent. Therefore CMB photons were freely traveling to the entire\nthe Universe. This process is called as decoupling. Then photons reached us from\nall direction from the last scattering surface1 [7, 22]. The temperature of the CMB\nis T = 2.725 ± 10−5K. The temperature anisotropies originate from metric per-\nturbations in the inflation phase of the Universe as seen in the previous chapter.\nThe CMB radiation is polarized because of the last scattering by free electrons\n(Thomson scattering) during decoupling. The polarization pattern can be decom-\nposed into two components: Curl-free component called ’E-Mode’ (electric-field)\nor ’gradient-mode’ and Grad-free component called ’B-Mode’ (magnetic-field) or\n’curl-mode’ [25, 119]. The B-Mode is impacted by gravity waves produced during\nthe inflation epoch at the early Universe. The CMB power spectrum depends on\ncosmological parameters. The high accuracy measurements the temperature and\npolarization anisotropies of the CMB allow us to measure the density of energy\ncomponent such as dark energy, dark matter, baryons. The table 3.1 shows the\nlatest Planck results for fiducial cosmological parameters of the ΛCDM (the Uni-\nverse has cosmological constant) concordant standard cosmological model which\nis described in chapter 2.\n\nIn 1964, Robert Wilson and Arno Penzias [92] first detected CMB by using a large\nradio antenna and they got the Nobel Prize in Physics in 1978. The first space\n1The imaging surface of a sphere which photons travel to us since the decoupling happened\naround at 3000 K\n\n\n\nThe Cosmic Microwave Background (CMB) 48\n\nParameter Value Description\nΩbh2 0.02237 ± 0.00015 Physical baryon density parameter\nΩch\n\n2 0.1200 ± 0.0012 Physical dark matter density parameter\nΩΛ 0.6847 ± 0.0073 Dark energy density parameter\nτ 0.0544 ± 0.0073 Reionization optical depth\nns 0.9649 ± 0.0042 Scalar spectral index\n\n109As 2.092 ± 0.034 Amplitude scalar of power spectrum\nH0 67.36 ± 0.54 Hubble constant\nΩb 0.0486 ± 0.0010 Baryon density parameter\nΩm 0.3153 ± 0.0073 Matter density parameter\nΩc 0.2589 ± 0.0057 Dark matter density parameter\n\nρc (kg/m3) (8.62± 0.12)× 10−27 Critical density\nAge/Gyr 13.797 ± 0.023 Age of the Universe\n\nσ8 0.8111 ± 0.0060 Fluctuation amplitude at 8h−1 Mpc\nNeff 3.00+0.57\n\n−0.53 Effective number of relativistic degrees of freedom∑\nmν 0.12 eV/c2 Sum of three neutrino masses (Planck + BAO)\n. . . . . . . . .\n\nTable 3.1: ΛCDM model, the fiducial cosmological parameters table from\nPlanck result 2018 results with 68 % confidence limits (CL) [111, 113, 116, 119].\n\nmission to detect CMB anisotropies is Cosmic Background Explorer (COBE).\nIn 1989 COBE was launched by NASA and placed into Sun-synchronous orbit2.\nCOBE measured the CMB temperature and showed that the CMB spectrum is a\nblack-body with a very high accuracy at 2.725 Kelvin. The team got the Nobel\nPrize in Physics in 2006. Studying these tiny fluctuations in more detail, other\nballoons and ground-based experiments after COBE such as: BOOMERanG ex-\nperiment reported that the highest power fluctuation occurs at around 1◦ in 2000,\nDegree Angular Scale Interferometers (DASI) experiment detected the polariza-\ntion of the CMB and the Cosmic Background Image (CBI) experiment measured\nthe E-mode, as well as BICEP, POLABEAR . . . The second generation space mis-\nsion, the Wilkinson Microwave Anisotropy Probe (WMAP) was launched in 2001.\nThe Planck mission was launched in 2009 to study the CMB with unprecedented\naccuracy. After the measurement of the tiny fluctuations of the CMB temperature\nby WMAP, the Planck instrument measured with high accuracy the temperature\npower spectrum and mapped the CMB as the whole sky. In addition, thanks to a\nbetter angular resolution and sensitivity, the Planck experiment gave very inter-\nesting constraints on primordial B-Modes. A join analysis of BICEP2, the Keck\narray and Planck data in 2015 set a limit on the tensor-to-scalar ratio of r < 0.12\n\n2It is geocentric orbit which has the same local mean solar time\n\n\n\nThe CMB 49\n\n[20]. The future CMB experiments aim to detect evidence of inflation Universe\nusing B-modes polarization.\n\nIn this chapter, I describe briefly statistic of the CMB, it is decomposition on the\nspherical harmonic space leading to the monopole, the dipole, multipoles and the\nangular power spectrum of the CMB temperature. I also describe the angular\nsize to the last scattering surface δθLS, the optical depth to the ionization epoch\nτ . On the one hand, I present the mechanism of CMB temperature anisotropies\nwhich can explain acoustic oscillations peaks in the temperature angular power\nspectrum. I describe also CMB photons polarization and its angular power spectra.\nIn addition, I describe briefly non-Gaussianity in the CMB, gravitational lensing,\nCMB spectral distortions. I describe the main foreground components as thermal\ndust, free-free emission, synchrotron emission, spinning dust, and main systematic\neffects as beam asymmetry, cosmic rays, 1/f noise, bandpass mismatch. I also\npresent the ground-based QUBIC experiment, the general principle as well as the\ninstrument, and the proposed space mission LiteBIRD.\n\n3.1 The CMB\n\nAfter ∼ 13.7 billions of years, the Universe has expanded and cooled, the wave-\nlength of the photons has stretched (redshift) into roughly 1 millimeter (we can\nsee the CMB on the old analog television snow at the level of ∼ 1%) and the\nCMB temperature has decreased to around T0 = 2.725 Kelvin. These photons\nfill everywhere in the Universe today and can be detected by far infrared and\nradio telescopes. The density number is nγ ≈ 400 photons per cubic centimeter\ncm−3 or 10 trillion photons per second per squared centimeter which is about 2\nbillion times the baryon density [71]. The CMB is anisotropic at the level of 10−3\n\ndue to the motion of our Solar system, and the primordial anisotropies are about\n10−5 ≡ δTCMB ≈ 30µK.\n\nThe CMB is a black body, then following a Planck function of frequency and\ntemperature:\n\nB(ν,T) =\n2hν3\n\nc2\n\n1\n\nehν/kBT − 1\n[Wm−2 sr−1 Hz−1]. (3.1)\n\n\n\nThe CMB 50\n\nThe energy density of radiation is\n\nεrad = αT 4, (3.2)\n\nwhere the radiation constant (or Stefan–Boltzmann constant):\n\nα =\nπ2k4\n\nB\n\n15~3c3\n= 7.565× 10−16 Jm−3K−4. (3.3)\n\nWe already know that ρrad ∝\n1\n\na4\n. Then, we have the relationship between tem-\n\nperature and scale factor equation.\n\nT ∝ 1\n\na\n. (3.4)\n\nFrom the equation 2.5 we have the equation of the redshift and the temperature\n\nTemitted = Tobserved(1 + z). (3.5)\n\nThe CMB temperature maps as observed by COBE and Planck mission are shown\nin figure 3.1.\n\nThe ratio of photon to baryon\n\nThe present energy density of photon is given in the equation 3.2 by:\n\nεrad(t0) = 4.17× 10−14 Jm−3. (3.6)\n\nThe energy of photons at temperature T= 2.725 K is\n\nErad ' 3kBT = 7.05× 10−4eV. (3.7)\n\nThe present number density of photons is calculated as:\n\nnγ =\nErad\nεrad(t0)\n\n= 3.7× 108 m−3. (3.8)\n\nWe can see that in a cubic metre, there are billion CMB photons. From the table\n3.1 we have the baryon density parameter and the critical density parameter. Then\nwe can calculate the baryon energy density\n\nεb = ρbc\n2 = Ωbρcc\n\n2 ' 3.77× 10−11 J m−3. (3.9)\n\n\n\nThe CMB 51\n\nFigure 3.1: From top to bottom, the CMB temperature monopole map\nTCMB ≈ 2.725K of the COBE Differential Microwave Radiometers (DMR), the\nCMB dipole map δTCMB = 3.3 mK and the CMB temperature anisotropies map\nis measured by the Planck satellite δTCMB/T ≈ 10−5. Credit: Planck/ESA team\n\n\n\nThe CMB 52\n\nIf we consider that the individual proton and neutron have rest mass about 939\nMeV, then we find the number density of baryons is\n\nnb = 0.25m−3. (3.10)\n\nThe ratio of nγ/nb ∼ 1.48×109, it means that the number of photons is around 1.5\nbillion times the number of baryons today. The Cosmic Microwave Background\nobserved light element abundances and give the constraint on the baryon density\nin the Universe.\n\n0.016 ≤ Ωbh\n2 ≤ 0.024 . (3.11)\n\nStatistical description of the CMB.\n\nWe can assume that the CMB temperature in a direction is described by a random\nGaussian field. Let us define the dimensionless CMB temperature anisotropies:\n\nΘ(n̂) ≡ δT\n\nT\n(θ, ϕ) =\n\nT(θ, ϕ)− T̄\n\nT̄\n. (3.12)\n\nWhere we denoted in a unit vector Θ(n̂) ≡ Θ(θ, ϕ). T (θ, ϕ) is the temperature\nin the sky direction (θ, ϕ), T̄ is the mean temperature. The CMB temperature\nanisotropies are decomposed in spherical harmonics basis:\n\nδT\n\nT\n(θ, ϕ) =\n\n∞∑\n\n`=1\n\n∑̀\n\nm=−`\na`mY`m(θ, ϕ) . (3.13)\n\nWith Y`m the spherical harmonic of degree ` and order m.\n\nY`m(θ, ϕ) = (−1)m\n\n√\n2`+ 1\n\n4\n\n(`−m)!\n\n(`+m)!\neimϕPm\n\n` (cos θ). (3.14)\n\nThis basis is adapted for the decomposition on the surface of a sphere, θ, ϕ repre-\nsent colatitude and longitude. Pm\n\n` is the associated Legendre polynomial.\n\nPm\n` (cos θ) =\n\n(−1)m\n\n2``!\n\n(\n1− cos2 θ\n\n)m/2 d`+m\n\nd cos`+m θ\n\n(\ncos2 θ − 1\n\n)`\n.\n\nP 1\n1 (cos θ) = sin θ,\n\nP 1\n2 (cos θ) = 3 cos θ sin θ,\n\nP 2\n2 (cos θ) = 3 sin2 θ. (3.15)\n\n\n\nThe CMB 53\n\nThe multipoles coefficient is can be expressed as:\n\na`m =\n\nˆ\nY ∗`m(θ, ϕ)\n\nδT\n\nT\n(θ, ϕ)dΩ . (3.16)\n\nWhere dΩ is the solid angle. This implies the following property of spherical\nharmonics.\n\nˆ\ndΩY`m(n̂)Y ∗`′m′(n̂) = δ``′δmm′ . (3.17)\n\nIf we sum over the multipole m which is related to orientation, we have multipole\nnumber function which is related to the angular size.\n\n∑\n\nm\n\n|Y`m(n̂)|2 =\n2`+ 1\n\n4π\n. (3.18)\n\nCMB monopole\n\nThe mean temperature of CMB is T = 2.725 K, and this is the monopole compo-\nnent of the CMB map.\n\nCMB Dipole ` = 1\n\nThe dipole pattern (hot and cold are opposite direction on the sky) in the CMB\nmap is dominated by the Doppler shift of the relative motion of the Solar system\nwith respect to the CMB rest frame.\n\nT = TCMB\n\n(\n1− v\n\nc\n\n2\n)1/2\n\n1− v\n\nc\ncos θ\n\n,\n\n= TCMB\n\n(\n1 +\n\nv\n\nc\ncos θ +\n\nv2\n\nc2\n\n(\ncos2 θ − 1\n\n2\n\n))\n. (3.19)\n\nThe first order of the dipole is related to the angle between the observer direction\nand the dipole axis [9, 134, 149].\n\nδT = TCMB\nv\n\nc\ncos θ = 3.37× 10−3 cos θK. (3.20)\n\nThe dipole measurement on the CMB map indicates that the Doppler shift ve-\nlocity of the Solar system is around 370 km/s with respect to the CMB frame.\nThe motion of the Earth around the Sun is around 30 km/s. This motion is an\nadditional dipole contribution. The effect of the Earth-Sun motion is very well\n\n\n\nThe CMB 54\n\nunderstood and allow to calibrate and monitor the gain as a function of time for\na single detector. The sky map includes only the average dipole. Normally the\nsignal of the dipole is removed for the anisotropies study.\n\nMultipoles moments `\n\nThe higher multipole moments are result of density and tensor perturbations of\nthe early Universe. Theoretically, a`m are described by a Gaussian random process\n[134]. The angular wavelength of the fluctuation is θ =\n\n180\n\n`\n, the temperature\n\nbetween points on the sky separated by angle θ. For example ` = 180 corresponds\nto about 1 degree on the sky. The angular resolution of the COBE satellite being\n7◦, then it can measure up to a resolution of ` ' 180/7 ' 26. Similarly the WMAP\nsatellite had 0.23◦ ⇒ ` ' 780, the Planck satellite had the angular resolution of 5\narcminutes, it allows to measure up to ` ' 2200.\n\nAngular power spectra\n\nSince the multipoles coefficient a`m represent a deviation from the average tem-\nperature, the mean value is zero (〈δT 〉 = 0).\n\n〈a`m〉 = 0, (3.21)\n\nThe variance C` of the coefficients a`m is called the angular power spectrum,\n\nC` ≡ 〈|a`m|2〉 =\n1\n\n2`+ 1\n\n∑\n\nm\n\n〈|a`m|2〉 . (3.22)\n\nC` is independent of m because of the isotropic nature of the random process. The\nm are represented to the orientation while the ` are represented the angular size\nof the anisotropy of orientation. The two-point covariance of a`m is calculated by\napplying equations 3.17 and 3.22.\n\n〈a`ma∗`′m′〉 =\n\nˆ\nY ∗`m(n̂)Y`′m′(n̂\n\n′)\n\n〈\nδT\n\nT\n(n̂)\n\nδT\n\nT\n(n̂′)\n\n〉\ndΩdΩ′\n\n= δ``′δmm′C`. (3.23)\n\nThe variance of the temperature anisotropies and the observed angular power\nspectrum are related to the multipoles `\n\n〈(\nδT\n\nT\n(n̂)\n\n)2\n〉\n\n=\n\n〈∑\n\n`m\n\na`mY`m(n̂)\n∑\n\n`′m′\n\na∗`′m′Y\n∗\n`′m′(n̂)\n\n〉\n\n\n\nThe CMB 55\n\n=\n∑\n\n``′\n\n∑\n\nmm′\n\nY`m(n̂)Y ∗`′m′(n̂) 〈a`ma∗`′m′〉\n\n=\n∑\n\n`\n\nC`\n∑\n\nm\n\n|Y`m(n̂)|2\n\n=\n∑\n\n`\n\n2`+ 1\n\n4π\nC`. (3.24)\n\nThe angular power spectrum depends on the power spectrum P(k) of density\nperturbations as well as CMB temperature anisotropies. If we have a uniform\nprimordial power spectrum in a logarithmic interval, and the inflation predicts\nthat the primordial power spectrum would be nearly a constant, then normally\n\nthe angular power spectrum is flat in the representation\n`(`+ 1)\n\n2π\nC`. The angular\n\npower spectrum plays an important role in the statistical analysis of the CMB.\nThe angular power spectrum provides the information on cosmological parameters\nas well as the early Universe scenario.\n\nThe cosmic variance is the squared difference between observed spectrum Ĉ` and\nthe theoretical spectrum C`, in case of noiseless observation:\n\n〈(\nĈ` − C`\n\n)2\n〉\n\n=\n2\n\n2`+ 1\nC2\n` . (3.25)\n\nThe cosmic variance is a fundamental limit of experimental measurement of the\nCMB to compare with theory. And it is important at low ` (the large scales).\nNote that a ground-based experiment can not cover full sky, therefore the cosmic\nvariance is a function of the inverse sky observation fobs [149]:\n\n〈(\nĈ` − C`\n\n)2\n〉\n\n=\n2\n\n(2`+ 1)fobs\nC2\n` . (3.26)\n\nThe two-point angular correlation function of the temperature on the sky is a\nfunction of cos θ = n̂1 · n̂2 [83]:\n\nC (cos θ) ≡ 〈T (n̂1)T (n̂2)〉 =\n∑\n\n`\n\n2`+ 1\n\n4π\nC`P` (cos θ) . (3.27)\n\nThe measured and theoritical CMB temperature angular power spectrum CTT\n`\n\nare plotted in figure 3.2. It contains three different regions [134, 149]:\n\n• The Sachs-Wolfe plateau ` ≤ 100: At the large scales the variation in\ngravitational potential and temperature fluctuations are statistically nearly\n\n\n\nThe CMB 56\n\nFigure 3.2: The temperature angular power spectrum\n`(`+ 1)\n\n2π\nCTT` measured\n\nby the Planck mission. The power spectrum has mainly three regions which\nare Sachs-Wolfe plateau; acoustic oscillation region and the damping. The red\ncurve is the predicted theoretical spectrum while blue dots are the Planck data\n\nbest-fit with 6 cosmological parameters. Credit:ESA/Planck team\n\nflat. It corresponds to mode that did not enter the horizon at the time of\nCMB emission.\n\n• Acoustic oscillations peaks 100 ≤ ` ≤ 1000: At the small scale, baryons-\nphotons fluid interacted by photons pressure and baryons inertia, and pro-\nduced oscillations in the CMB spectrum today.\n\n• Damping tail ` ≥ 1000: At the very small scale, the smoothed damp tail\nis produced by the diffusion of photons-baryons fluid and imperfect recom-\nbination processes.\n\nThe first peak has angular scale ∼ 1◦ ≡ ` ∼ 200. This scale is related to the\ngeometry of the Universe. The second peak is lower than the first peak, the ratio\nof the second to the first peak tells us about the baryon density. 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sha1_base64=\"jPg7a/HOKbhKd4JdW/+Kih4wZh8=\">AAACFnicbVA9SwNBEN3z2/gVtRRkUQQtDHdaaBkQxFIhH0IuhL29uWR1b+/YnRPCcT/A2kZrf4WNhSK2Yue/cZNY+PVg4PHeDDPzglQKg6774YyNT0xOTc/MlubmFxaXyssrDZNkmkOdJzLR5wEzIIWCOgqUcJ5qYHEgoRlcHg385hVoIxJVw34K7Zh1lYgEZ2ilTnnXlxDhNvXDSDOe+yFIZLRGi7xWUF+Lbg93OvmFXaegX3TKm27FHYL+Jd4X2azu9+7Xi7vr00753Q8TnsWgkEtmTMtzU2znTKPgEoqSnxlIGb9kXWhZqlgMpp0P3yrollVCGiXalkI6VL9P5Cw2ph8HtjNm2DO/vYH4n9fKMDps50KlGYLio0VRJikmdJARDYUGjrJvCeNa2Fsp7zGbD9okSzYE7/fLf0ljr+K5Fe/MpnFMRpgha2SDbBOPHJAqOSGnpE44uSEP5Ik8O7fOo/PivI5ax5yvmVXyA87bJ6iKorc=</latexit><latexit sha1_base64=\"SlMim8GWg/IoE9T0LqQlz7M8ocE=\">AAACFnicbVDJSgNBEO1xjXGLevTSGAQ9GGa86DEgiMcI2SATQk9PTdLa0zN01whhmK/w4q948aCIV/Hm39hZDm4PCh7vVVFVL0ilMOi6n87C4tLyympprby+sbm1XdnZbZsk0xxaPJGJ7gbMgBQKWihQQjfVwOJAQie4vZj4nTvQRiSqieMU+jEbKhEJztBKg8qJLyHCI+qHkWY890OQyGiTFnmzoL4WwxEeD/Ibu07BuBhUqm7NnYL+Jd6cVMkcjUHlww8TnsWgkEtmTM9zU+znTKPgEoqynxlIGb9lQ+hZqlgMpp9P3yrooVVCGiXalkI6Vb9P5Cw2ZhwHtjNmODK/vYn4n9fLMDrv50KlGYLis0VRJikmdJIRDYUGjnJsCeNa2FspHzGbD9okyzYE7/fLf0n7tOa5Ne/ardYv53GUyD45IEfEI2ekTq5Ig7QIJ/fkkTyTF+fBeXJenbdZ64Izn9kjP+C8fwFRRJ98</latexit>\n\nFigure 3.3: The surface of last scattering (LS).\n\nAngular size δθLS to the last scattering surface\n\nThe CMB temperature fluctuations at the time of last scattering have the small\nangular size δθ related to the physical size λphys on the surface of last scatterings,\nand to the comoving angular diameter distance dA(z) to the last scattering surface,\nsuch as:\n\nδθLS =\nλphys\ndA(z)\n\n. (3.28)\n\nThis is illustrated in figure 3.3. The comoving angular diameter distance redshift\nis related to the horizon distance of two regions on the last scattering surface\ndhor(tLS):\n\ndhor(tLS) ≡ dA(z) =\n\nˆ zLS\n\n0\n\ncdz\n\nH(z)\n. (3.29)\n\nThe Hubble parameter is given by the equation 2.44,\n\nH(z) = H0\n\n[\nΩm;0(1 + z)3 + Ωr;0(1 + z)4 + Ωk;0(1 + z)2 + ΩΛ;0\n\n]1/2\n.\n\nFor a flat Universe, the comoving angular diameter distance and comoving distance\nare equal. For matter dominated Universe, H(z) ≈ (1 + z)3/2. Then:\n\ndA(z) =\nc\n\nH0\n\nˆ zLS\n\n0\n\n(1 + z)−3/2dz =\n2c\n\nH0\n\n(\n1− (1 + zLS)−1/2\n\n)\n. (3.30)\n\n\n\nThe CMB 58\n\nBecause at the last scattering surface zLS ∼ 1100, the comoving angular diameter\ndistance redshift is approximated as\n\ndA(z) =\n2c\n\nH0\n\n. (3.31)\n\nThe physical distance λphys at the last scattering surface is the comoving distance\nor the particle horizon length dH(z):\n\nλphys ≡ dH(z) =\n\nˆ ∞\nzLS\n\ncdz\n\nH(z)\n=\n\n2c\n\nH0\n\n(1 + zLS)−1/2. (3.32)\n\nThe angular size is calculated by dividing result of the equation 3.32 to the equation\n3.31.\n\nδθLS = (1 + zLS)−1/2 ≈ 1.7◦ . (3.33)\n\nIt means that the scale angle larger than 1.7◦ (` ' 105) were not in contact\nat the surface of the last scattering. This also refers to the horizon problem of\nthe Universe which we mentioned in the chapter 2 section 2.6, and is solved by\ninflation.\n\nOptical depth τ\n\nAfter CMB emission, first stars and galaxies reionize the Universe. The reioniza-\ntion epoch happens at the redshift around z = 6− 25 as illustrated in figure 2.2.\nThe reionization epoch provides information of first stars and galaxies formation.\nThe reionization epoch is constrained by the observed CMB. The optical depth\nof reionization refers to CMB photons last scattered from free electrons in the\nintergalactic medium. The Thomson scattering is an interaction of photons with\nelectrons.\n\nγ + e− → γ + e−. (3.34)\n\nThe differential cross-section of the interaction in a solid angle dΩ is [155]:\n\ndσe\ndΩ\n\n=\n3σT\n8π\n|ε · ε′|2. (3.35)\n\nwhere σT = 6.65 × 10−29m2 is the Thomson cross-section. |ε · ε′| is the angle\nbetween scattered and incident photons. The mean free path (the distance of\ntraveling photons to electrons) is defined as [129]:\n\nλγ =\n1\n\nneσe\n, (3.36)\n\n\n\nCMB temperature anisotropies 59\n\nwhere ne is the number density of electrons, the number density of electrons de-\npends on the redshift ne(z). Since the speed of photons is c, the scattering rate of\nphotons is given by\n\nΓ(z) =\nc\n\nλγ\n= ne(z)σec, (3.37)\n\nWhen we collect CMB photons at time t0, these photons have been scattred by\nfree electrons in the intergalactic medium. Thus the optical depth [129] is:\n\nτ(t) =\n\nˆ t0\n\nt\n\nΓ(t′)dt′, (3.38)\n\nat the surface of last scattering, τ = 1. We can change the variable into the scale\nfactor and remembering that H = ȧ/a:\n\nτ(a) =\n\nˆ 1\n\na\n\nΓ(a)\nda\n\nȧ\n=\n\nˆ 1\n\na\n\nΓ(a)\n\nH(a)\n\nda\n\na\n. (3.39)\n\nWe can express the optical depth as a function of redshift using 1 + z = 1/a:\n\nτ(z) =\n\nˆ z\n\n0\n\nΓ(z)\n\nH(z)\n\ndz\n\n1 + z\n=\n\nˆ z\n\n0\n\ndz\n\nH(z)(1 + z)\nne(z)σec . (3.40)\n\nThe Hubble parameter is given as a function of redshift in the equation 2.44.\nAccording to the Beer-Lambert law of attenuation of light traveling in the ma-\nterial and physics of CMB photons, the amplitude of temperature angular power\nspectrum at the reionization bump is approximately\n\nC` ≈ Ase\n−2τ , (3.41)\n\nwhere As is the scalar amplitude. The measurement of τ value plays an im-\nportant role to determine the primordial B-mode power spectrum as well as the\ntensor-to-scalar value and to study the epoch of reionization. On the other hand,\nthe Sunyaev-Zeldovich effect which is distortions of CMB photons through the\nInterGalactic Medium (IGM) by interaction with high energy electrons (inverse\nCompton scattering), effects the temperature fluctuations on all scales [91]:\n\nδT\n\nT\n→ δT\n\nT\ne−τ . (3.42)\n\n\n\nCMB temperature anisotropies 60\n\n3.2 Physics of CMB temperature anisotropies\n\nMany physical sources contribute to CMB anisotropies. We classify them into\nprimary and secondary anisotropies. The primary anisotropies arise at the recom-\nbination epoch while the secondary anisotropies are caused by the anisotropies\nof photons distribution between the last scatering surface and the observers. As\nwe already described, the early Universe was filled by baryons-photons fluid in a\ngravitational potential interaction, baryons matter attract each other and fall in\nthe dark matter potential well. When the baryons density increases, the pressure\nwill be increase then the repulsion force due to radiation pressure appears. There\nare some of the primary anisotropies sources which described the mechanism of\ntemperature anisotropies on large, intermediate and small angular scale [7, 91]:\n\n• The Sachs-Wolfe (SW) effect [130]: Gravitational perturbations lead\nthe intrinsic temperature variations at the last scattering surface due to\nphotons climbing out or fall into gravitational potential wells δΦ. There\nis also effect of higher temperature in higher density. This process links\ntemperature anisotropies and gravitational potential fluctuations in the early\nUniverse. The Sach-Wolfe effect is the combination of two effects arising from\ngravitational potential perturbations. Firstly, gravitational redshift effect\ndue to photons climb out or fall into of the gravitational potential wells.\nThe gravitational redshift is determined by:\n\nδν\n\nν\n≈ δΦ\n\nc2\n, (3.43)\n\nwhere ν is the frequency of the photons. Secondly, the time dilation at the\nlast scattering surface, it occurs at the higher temperature at an overdensity\nregion. The gravitational redshift effect is caused of the time dilation of\nscatering photons δt/t = δΦ/c2. Remember that the CMB temperature\nT ∝ 1/a(t), and in the matter-domonated era nγ ∝ ρm ∝ a−3 ∝ T 3, it\nfollows that δρ/ρ = 3δT/T . Moreover we can imply also a ≈ t2/3, ρ ∝ t−2.\nFinally we get [52, 148]:\n\n1\n\n3\n\nδρ\n\nρ\n=\n−2\n\n3\n\nδt\n\nt\n=\n\n2\n\n3\n\nδν\n\nν\n≈ −2\n\n3\n\nδΦ\n\nc2\n. (3.44)\n\n\n\nCMB temperature anisotropies 61\n\nThe combining two processes are Sach-Wolfe effect, these fluctuations dom-\ninate at the large scale [126, 130]:\n\nδT\n\nT\n≈ 1\n\n3\n\nδΦ\n\nc2\n. (3.45)\n\n• Density (Adiabatic) perturbations: The tight coupling of baryonic mat-\nter and radiations can compress the radiation resulting on the increasing of\nthe temperature. The fraction of temperature perturbations in the radiation\nis as same as the fraction of density perturbations.\n\n(\nδT\n\nT\n\n)\n\nobs\n\n=\nδρ\n\nρ\n. (3.46)\n\n• Doppler effect: The Doppler velocity perturbations, there has a variation\nin wavelength of the photons when they are emitted by measuring structures\nwith respect to us. If v is the photon fluid velocity in the last scattering\nsurface and r̂ is the direction along the line of sight:\n\nδT\n\nT\n=\nδv · r̂\nc\n\n. (3.47)\n\nFigure 3.4: The physical mechanism of CMB temperature anisotropies. SZ\nstands for the Sunyaev-Zeldovich effect, ISW stands for the Integrated Sachs-\nWolfe Effect. The desity perturbation at last scattering surface can be analysed\n\nin Fourier space with modes of wavevector k [91].\n\nA full treatment solution of the mechanism of temperature anisoptropies was de-\nscribed in Wane Hu thesis in 1995 [51, 54, 56]. The mechanism of the original\n\n\n\nCMB polarization 62\n\nanisotropies from baryons and gravity is that gravity introduced potential wells\nand affected to the amplitude of temperature oscillations. Baryons-photons per-\nturbations oscillated the process created acoustic peaks in the temperature fluc-\ntuations by compression and expansion of photons-baryons plasma fluid. During\ntheir travel, photons interacted with the large-scale structures of the Universe\nfrom the last scattering surface. Consequently, secondary anisotropies are present\nin the CMB maps, and then these oscillation peaks depend on the dark matter,\nthe dark energy, the baryons density and so on. Figure 3.5 shows the shift of the\ntemperature angular power spectrum with respect to cosmological parameters.\n\nFigure 3.5: The variation of Baryon acoustic peaks of the temperature angular\npower spectrum and 4 cosmological parameters which are varied around the\nfiducial point: The curvature Ωtot = 1, the cosmological constant or dark energy\nΩΛ = 0.65, the physical baryon density Ωmh\n\n2 = 0.02, the physical matter density\nΩmh\n\n2 = 0.147 [52].\n\n\n\nCMB polarization 63\n\n3.3 CMB polarization\n\nTemperature anisotropies at the last scattering surface are evidence of primordial\nfluctuations at the early Universe. Local temperature anisotropies at the last\nscattering surface lead to the polarization of photons via Thomson scattering as\nshown in figure 3.6.\n\ne-\n\nThomson  \nScattering\n\nQuadrupole \nAnisotropy\n\nLinear  \nPolarization\n\n𝜸’\n\n𝜸’\n\n𝜸\n\nFigure 3.6: The temperature perturbation via quadrupole anisotropies with\nThomson scattering of hot, cold radiations and electrons generated linear hori-\n\nzontal/vertical polarization.\n\nThomson scattering is the interaction of an electromagnetic wave with a free elec-\ntron. The scattered wave is polarized perpendicular to the incidence direction. Be-\ncause of incident photons from a perpendicular direction have different intensities,\nthen the result is linearly polarized. Local quadrupole anisotropies in the distri-\nbution on photons produces a net polarization (because the poles of an anisotropy\nare 360◦/4 = 90◦) (` = 2, m = 0,±1,±2). The CMB photons only scatter when\nthere are still free electrons in the last scattering surface. Thus the polarization\ncould be produced during a short time at the end of recombination epoch. Conse-\nquently, we only have a small fraction of polarization of CMB photons. It depends\non the thickness of the last scattering duration. The polarized signal is at about\n10−6 or several µK which is ∼ 10% of the temperature anisotropies of 10−5 level.\nThis order of magnitude is the reason for experimental challenges [57]. These\nquadrupole anisotropies are created by scalar and tensor perturbations.\n\n\n\nCMB polarization 64\n\nm=0\n\nv\n\nScalars \n(Compression)\n\nhot\n\nhot\n\ncold\n\ntrough\n\ncrest\n\nm=1\n\nv\n\nVectors \n(Vorticity)\n\ncrest\n\ntrough\n\ntrough\n\nm=2 \n \n\nTensors \n(Gravity Waves)\n\nFigure 3.7: There have three different sources of quadrupoles anisotropies.\nScalar perturbations originated from density perturbations, vector perturbations\noriginated from the primordial fluid. In the physic of inflation, these vector per-\nturbations are decay with the expansion of the Universe then we are negligible\nthem. Tensor perturbations are evidenced by primordial gravitational waves\nfrom inflation which affected to density fluctuations resulting from the transfor-\n\nmation of gravitational well [57, 149]. credit: Wayne Hu\n\nIf a monochromatic light wave propagates in the z−direction, the electric field at\na given point:\n\nEx = E0x cos (ω0t− ϑx(t)) ; Ey = E0y cos (ω0t− ϑy(t)) , (3.48)\n\nIn order to simplify understanding of the polarization properties, the polarization\nof the CMB can be described by the Stokes parameters. The Stoke parameters\nare defined as:\n\nI ≡ 〈E2\n0x〉+ 〈E2\n\n0y〉;\nQ ≡ 〈E2\n\n0x〉 − 〈E2\n0y〉;\n\nU ≡ 〈2E0xE0y cos(ϑ)〉;\nV ≡ 〈2E0xE0y sin(ϑ)〉. (3.49)\n\nWhere ϑ = ϑy − ϑx, Eox and Eoy are components orthogonal to the propagation.\nThe Stoke parameter I represents for intensity of the radiation which can be the\nCMB temperature. The linear polarization is Q and U parameters, Q parameter\nstates in x-y directions while U parameter is rotated by 45◦. The V parameter\nis the circular-polarization, in the Thomson scattering process, V is vanished.\nNature light or unpolarized radiation has Q = U = V = 0. If the CMB radiation\nis rotated during its propagation because of Faraday rotation with a rotated angle\n\n\n\nCMB polarization 65\n\nα, we can write:\n\n(\nQ′\n\nU ′\n\n)\n=\n\n(\ncos 2α sin 2α\n\n− sin 2α cos 2α\n\n)(\nQ\n\nU\n\n)\n. (3.50)\n\nThe linear polarization parameters Q, U can be described as a symmetric trace-free\n2× 2 tensor, and depend on the coordinate system. Nevertheless it is possible to\ndecompose the polarization Q, U in 2 fields which are independent of the coordinate\nsystem. We can decompose the polarization Q, U in the second-order spin spherical\nharmonics space are [144]:\n\n(Q± iU) (n̂) =\n∑\n\n`m\n\na±2`m Y±2`m(n̂). (3.51)\n\nWe can then define the two orthogonal combination of E-mode and B-mode by\nanalogy to the to electric (also known as Gradient-G) and magnetic (also known\nas Curl-C) fields. Their spherical harmonics coefficents are introduced as [24, 135,\n153, 154]:\n\naE`m = −a2`m + a−2`m\n\n2\n\naB`m = i\na2`m − a−2`m\n\n2\n. (3.52)\n\nThen the two scalar (spin 0) E and B are defined as\n\nE(n̂) =\n∑\n\n`m\n\naE`mY`m(n̂)\n\nB(n̂) =\n∑\n\n`m\n\naB`mY`m(n̂). (3.53)\n\nThe E and B modes completely describe the linear polarization, E-mode polar-\nization is represented as curl-free modes which are radial around cold spots and\ntangential around hot spots. B-mode polarization is a divergence-free which has\ncurl-free also. B-mode polarization has vorticity around spots. Figure 3.8 shows\nthe pure maps of E- and B- mode patterns. They are invariant under coordi-\nnate rotations but in a parity transformation, the E-mode is unchanged while the\nB-mode changes sign. Scalar perturbations only produce E-mode, vector pertur-\nbations produce B-mode but they are sub-dominated because of the expansion\nof the Universe, therefore, believed to be negligible, tensor perturbations produce\nboth E-modes and B-modes thus detection of B-mode polarization is the evidence\n\n\n\nCMB polarization 66\n\nof primordial gravitational waves of the inflation epoch [18, 24, 136, 153].\n\nE Mode B Mode\nFigure 3.8: The representation of pure E- and B-modes.\n\nAs for equation 3.22, the angular power spectra of pure polarization, and temper-\nature cross polarization are defined as:\n\nCEE\n` =\n\n〈\n|aE`m|2\n\n〉\n\nCBB\n` =\n\n〈\n|aB`m|2\n\n〉\n\nCTE\n` =\n\n〈\naT`ma\n\n∗E\n`m\n\n〉\n\nCTB\n` =\n\n〈\naT`ma\n\n∗B\n`m\n\n〉\n\nCEB\n` =\n\n〈\naE`ma\n\n∗B\n`m\n\n〉\n. (3.54)\n\nThe second-order spin spherical harmonics have CTB\n` , CEB\n\n` equal 0. Figure 3.9 plot-\nted the theoretical angular power spectra of temperature, polarization and cross\nterm. The temperature angular power spectrum is well understood nowadays.\nThe E-mode polarization is measured by many experiments while the amplitude\nof B-mode is not yet known.\n\nAs discussions in the chapter 2 section 2.10.2 the scalar perturbations produce E-\nmodes polarization due to density perturbations while gravitational waves space-\ntime produces B-mode. The definition of h+ and h× depend on coordinates system,\nbut definitions of E-mode and B-mode polarisation do not depend on coordinates\nsystem. Therefore, h+ does not always give E, h× does not always give B. The\nimportant point is that h+ and h× always coexist. When a linear combination\n\n\n\nCMB polarization 67\n\nFigure 3.9: The temperature, polarization and temperature cross polarization\nangular power spectra. The 1σ statistical errors of the Planck satellite are shown\n\nby color boxed [52].\n\nof h+ and h× produces E, another combination produces B. At the small angular\nscale, gravitational lensing produces B-mode from E-mode. The amplitude of B-\nmode at large angular scales is related to the tensor-to-scalar ratio r which gives\nthe information of inflation scenarios, the energy of inflation in the early Universe\nas shown in equation 2.115 [57].\n\nWe can compute the power spectra in the Fourier space with the helppf spheri-\ncal Bessel transfer functions ∆T`(k),∆E`(k),∆B`(k). The details of the function\ncan be found in the [Seljak & Zaldarriaga, Kamionkowski] papers [64, 136, 153].\nThe angular power spectrum of polarization E-mode and the cross term TE is\ndominated by scalar perturbation in inflation, it means:\n\nCTE\n` ≈ (4π)2\n\nˆ\nk2dk PR(k)︸ ︷︷ ︸\n\ninflation\n\n∆T`(k)∆E`(k),\n\nCEE\n` ≈ (4π)2\n\nˆ\nk2dk PR(k)︸ ︷︷ ︸\n\ninflation\n\n∆2\nE`(k). (3.55)\n\n\n\nPrimordial non-Gaussianitiy in the CMB 68\n\nFor B-mode which are only created by tensors perturbation in inflation:\n\nCBB\n` = (4π)2\n\nˆ\nk2dk Ph(k)︸ ︷︷ ︸\n\ninflation\n\n∆2\nB`(k). (3.56)\n\nThe measurement of CBB\n` is the unique way to access tensor perturbation infor-\n\nmation.\n\n3.4 Primordial non-Gaussianity in the CMB\n\nThe inflation scenarios predict that the amplitude and the phase of energy density\nfluctuations in the early Universe are random variables following a very nearly\nGaussian statistics, small deviations from Gaussianity might be expected for same\ninflationary models. These deviations are called non-Gaussianity (NG). Therefore,\nthe statistical study of non-Gaussianity will tell us about the different theoretical\nmodels of inflationary paradigm. So far we have studied the second order statis-\ntics of the angular power spectrum of CMB fluctuations which depend on the\nprimordial power spectrum PR(k) in the Fourier space.\n\nThe primordial power spectrum contains all the information for a Gaussianity\nfield. We need to go to higher order statistics to study the non-Gaussianity, the\nnext order being the three-point function giving the primary measurement of non-\nGaussianity. The computation of non-Gaussianity requires careful expansion of\nthe third order of the action. Therefore I just show the main physical definition.\nThe bispectrum of 3-point function in Fourier space is:\n\n〈Rk1Rk2Rk3〉 = (2π)3δ(k1 + k2 + k3)BR(k1,k2,k3). (3.57)\n\nBR(k1,k2,k3) is the primordial bispectrum, it has degree −6 [18], the primordial\nbispectrum leaves a signature in the CMB angular bispectrum on the sphere is\n\nB`1`2`3\nm1m2m3\n\n= 〈a`1m1a`2m2a`3m3〉 . (3.58)\n\nThe local non-Gaussianity Rx is defined through the curvature perturbation around\nthe Gaussian Rg using a Taylor expansion. The local primordial non-Gaussianity\nis parametrized by the non-linear constant parameter fNL which quantifies the\n\n\n\nCMB spectral distortions 69\n\namplitude of non-Gaussianity for different shapes [18]:\n\nRx = Rg +\n3\n\n5\nfNL\n\n[\nRg(x)2 −\n\n〈\nRg(x)2\n\n〉]\n. (3.59)\n\nThe bispectrum of the local non-Gaussianity is\n\nBR(k1, k2, k3) =\n6\n\n5\nfNL [PR(k1)PR(k2) + PR(k2)PR(k3) + PR(k3)PR(k1)] . (3.60)\n\nThe experimental measurements set constraints on the fNL parameter. In addi-\ntion, the gravitational lensing of CMB also produced a level of non-Gaussianity,\nhence the study of non-Gaussianity is important to test the fundamental inflation\nphysics.\n\n3.5 Gravitational lensing\n\nThe CMB photons propagation is affected by the gravitational field of the large-\nscale structure, galaxies, and stars. Gravitational lensing was studied by Ein-\nstein in 1912 and also predicted by the general relativity theory. By generating\nspacetime curvature bright object can induce blending the propagation of light.\nMicrolensing is the lensing effect of stars in a galaxy, this effect is a useful tool\nto probe small scale substructures as well as dark matter in a galaxy, dark mat-\nter halos, while cosmic shear is produced by gravitational lensing at large-scale\nstructure of the Universe.\n\nThe CMB photons travel through the large-scale structure to us from the last\nscattering surface. Like cosmic shear of galaxy clusters, the temperature and\npolarization anisotropies of CMB is a type of the gravitational lensing effect and\nit distorts the hot and cold spots. In contrast to the cosmic shear, CMB lensing\nis the lensing of Gaussian random fields. The weak lensing effect converts E mode\nto B mode at small angular scales ` ≥ 300 and gravitational weak lensing has\ncontamination of B mode from primordial gravitational waves. We can calculate\nthe lensing effect in the CMB power spectrum, as well as the mixing of E-mode\nand B-mode polarization. Many studies of gravitational lensing in the CMB can\nbe found in the [7, 53, 55, 133, 138].\n\n\n\nCMB spectral distortions 70\n\nFigure 3.10: An example of CMB lensing effect on 10◦ × 10◦. Credit: Wayne\nHu & Takemi Okamoto.\n\n3.6 CMB spectral distortions\n\nCMB spectral distortions are departure the CMB spectrum from a pure black body\nspectrum due to its interaction with the matter in the Universe. Many processes\nare sources of spectral distortions such as reionization and structure formation,\ndecaying or annihilating particles, dissipation of primordial density fluctuations,\ncosmic string, primordial black hole, small-scale magnetic fields, adiabatic cooling\nof matter, cosmological recombination . . . [27]. The spectral distortions reach the\nlevel of∼ 10−7−10−6 relative to the CMB. There are two types of the CMB spectral\ndistortions which are the chemical potential µ ≤ 9× 10−5 and Compton y ≤ 1.5×\n10−5 distortions as shown in figure 3.11. µ distortions is created by photon injection\ndue to particle decays via Compton scattering at the early Universe [28, 143].\ny distortions is also named the thermal Sunyaev-Zel’dovich effect. The CMB\nphotons travel through galaxy clusters thus CMB photons can be scattered by\nhot electrons in the gas via Compton scattering, this phenomenon is so-called the\nthermal Sunyaev-Zel’dovich (SZ) effect [139, 140]. Consequently, the temperature\nof CMB photons is changed in the blackbody spectrum. The Compton y parameter\nis calculated as [29]:\n\ny =\n\nˆ\nkTe\nmec2\n\nneσTdl, (3.61)\n\nhere σT is Thomson cross section, ne and Te are the electron number density and\ntemperature respectively, dl is the path length through clusters. Basically, we have\nkTe/mec\n\n2 ∼ 0.01 and the Thomson optical depths τT =\n´\nneσTdl ∼ 0.01. Thus the\n\n\n\nForeground components 71\n\nvalue of y distortion is typically of the order of 10−4. The change in temperature\nof CMB photons due to thermal SZ effect is [29]:\n\n∆I(ν, n̂)\n\nICMB(ν, n̂)\n= y\n\nxex\n\nex − 1\n\n[\nx\nex + 1\n\nex − 1\n− 4\n\n]\n, (3.62)\n\nwhere x =\nhν\n\nkTCMB\n\n.\n\nPhoton energy Photon energy Photon energyPhoton energy\n\nD\nist\n\nor\ntio\n\nn \nSi\n\ngn\nal\n\nPhoton energy Photon energy Photon energy Photon energy\n\nPhoton energyPhoton energyPhoton energyPhoton energy\ny-distortion µ-distortion temperature shifty+µ+residual distortion\n\nRedshift104 2 x 1063 x 105103\n\nTime 2 months8 years7,000 years380,000 years\n\nH\nyd\n\nro\nge\n\nn\n l\n\nin\nes\n\nN\neu\n\ntr\na\n\nl \nH\n\nel\niu\n\nm\n l\n\nin\nes\n\nIo\nn\n\niz\ned\n\n H\nel\n\niu\nm\n\n l\nin\n\nes\n\nLa\nst\n\n S\nca\n\ntt\ner\n\nin\ng \n\nSu\nrf\n\na\nce\n\ntime-dependent information\n\nfull thermalizationscattering efficientscattering inefficient intermediate regime\n\nRecombination signal\n\nLy\nm\n\nan\n-α\n\nBa\nlm\n\ner\n-α\n\nPa\nsc\n\nhe\nn-\nα\n\nDistortion ≃10-7-10-6 \nrelative to blackbody\n\nMaximum of \nCMB blackbody\n\nMaximum of \nCMB blackbody\n\nBlackbody era\nDistortion \nvisibility\n\n5 x 106\n\n33\n%\n\n4% 0.\n2%\n\n0.\n00\n\n3%\n\nRecombination era\n\nR\nei\n\no\nn\n\niz\na\n\nti\no\n\nn\n\nB\nig\n\n B\na\n\nn\ng \n\nN\nu\n\ncl\neo\n\nsy\nn\n\nth\nes\n\nis\n\nSilk & Chluba, Science, 2014\n\nFigure 3.11: Different signals of CMB spectral distortions in the early Uni-\nverse. µ and y distortions are redshift independent. After the BigBang Nucle-\nosynthesis, the energy is release at redshift at ∼ 5× 106. µ distortions arises at\n3× 105 ≤ z ≤ 2× 106. The transition µ, y distortions is at 104 ≤ z ≤ 3× 105. y\ndistortion arises at recombination epoch due to atomic transitions of hydrogen\n\nand helium. Credit: Silk & Chluba\n\nFrom the study of spectral distortions, we can learn the reionization and structure\nformation history, the cosmological recombination spectrum, the dissipation of\nsmall scale acoustic modes, constraints on the tensor-to-scalar ratio B-modes and\nvarious scenarios [27].\n\n\n\nForeground components 72\n\n3.7 Foreground components\n\nIn order to estimate the CMB signal, we have to perform foreground component\nseparation. There are many sources of foregrounds which consists of emissions\nbetween us and the CMB. The most important are Glactic dust emission and\nGalactic synchrotron. Figure 3.12 shows the instrumental noise, the galactic fore-\nground, the galaxie cluster SZ emission. The transmission bands of future CMB\nmissions are expected to avoid the rotational lines of carbon monoxide which is\nemitted in the epoch of star formation in the Universe. CO lines: CO J = 1 → 0\n( 115.3 GHz), CO J = 2 → 1 ( 230.5 GHz), CO J = 3 → 2 ( 345.8 GHz), CO J =\n4 → 3 ( 461.0 GHz), CO J = 5 → 4 ( 576.3 GHz), CO J = 6 → 5 ( 691.5 GHz),\nCO J = 7→ 6 ( 806.7 GHz), J is the angular-momentum quantum number [128].\n\nFigure 3.12: The foreground components, the detector noise, galactic emission:\nDust, Free-free, synchrotron, cluster: Doppler effect, thermal SZ effect.\n\nFigure 3.13 indicates the dominated source of galactic components at frequencies\nrelevant to CMB observation. Foreground emissions are polarized, the B-mode\nsignal has ≤ 1 % of the level of the foreground emission. The sum of the foreground\nsignal is always larger than the CMB B-mode signal at all frequencies. So that\nthe foreground subtraction and separation play an important role in data analysis\nat different frequencies for CMB measurements [60, 106]. In this small section I\nfocus on the contamination of our galactic foreground emissions which are free-free,\nspinning dust, synchrotron and thermal dust.\n\n\n\nForeground components 73\n\nFigure 3.13: The microwave sky components, the frequency bands are the\nPlanck focal plane [105].\n\nThere have several component separation methods classified as internal template\nfitting, parametric methods, and non-parametric methods (internal linear combi-\nnation, independent component analysis). Commander is a Bayesian procedure\nfor the CMB and foreground components and Monte Carlo method used Gibbs\nsampling for CMB power spectrum [42]. NILC (Needlet Internal Linear Com-\n\nbination) for which, the CMB component is computed minimizing variance by lin-\near combination while the foreground is removed out using multi-frequency infarc-\ntion [37]. SEVEM (Spectral Estimation Via Expectation Maximisation)\n\nis an internal template subtraction method. The method estimates foreground\ncomponents using high and low frequency channels, then subtracting them out at\nCMB frequencies [44], SMICA (Spectral Matching Independent Compo-\n\nnent Analysis) is a method of the independent component analysis applied in\nthe angular spectrum domain [36, 60, 112]. These methods normally assume that\nobserved maps are a linear mixture of unknown components [127]. The meth-\nods assume the sky temperature at a pixel position n̂ and the frequency ν is a\n\n\n\nForeground components 74\n\nsuperposition of components Xi(n̂) and noise n(n̂, ν):\n\nT (n̂, ν) =\n∑\n\ni\n\nαi(ν)Xi(n̂) + n(n̂, ν), (3.63)\n\nhere αi(ν) are the component amplitude coefficients which depend on frequency\nof foreground emission.\n\n3.7.1 Thermal dust\n\nOne of the major foreground component effecting to the study of CMB is the\ngalactic thermal dust emission which arises from the interstellar dust grain in the\nmicrowave sky at ≥ 70GHz frequencies [101] and it is heated by stars light. The\nspectrum of thermal dust is a modified black-body\n\nIdust (n̂) = τν0 (n̂)×\n(\nν\n\nν0\n\n)βd\n× B(ν,Td). (3.64)\n\nHere τν0 (n̂) is the dust optical depth at frequency ν0, orientation n̂, βd is the spec-\ntral index. B(ν, T ) is the Planck function at frequency ν and the dust temperature\nTd [96]. The spectral index of dust is an important parameter of separation the\ndust and the CMB polarization [115]. Figure 3.14 represents the Planck 2018 re-\n\nPComm\nd\n\n3 300uKRJ at 353 GHz\n\nFigure 3.14: The Planck 2018 result of thermal dust polarization at 353 GHz\nusing Commander method with 5′ FWHM resolution [117].\n\nsult of the galactic thermal dust at 353 GHz. In the Galactic plane regions the\ndust temperature gradient can be observed from 14 K to 19 K. Because of differ-\nences in size and component then the emitted photons from thermal dust can be\n\n\n\nForeground components 75\n\npolarized. Observations have showed that the level of thermal dust polarization is\nabout 10 % [34, 114].\n\n3.7.2 Synchrotron\n\nSynchrotron radiation or magnetobremsstrahlung is emitted due to free electrons\ncosmic rays spiraling in the magnetic field of the Milky Way. The contamination\nof synchrotron is 10 % polarization of CMB signal at low frequencies (≤ 80GHz),\nit gets higher at lower frequency for the P06 mask |b| < 5◦ [60]. The intensity\nand the energy depend on the electron density and the strength of the magnetic\nfield, the energy distribution of electrons follows a power law Ne(E) ∝ E−p with\nindex p and the spectrum of synchrotron emission is:\n\nTsync (ν) ∝ B(p+1)/2νβs . (3.65)\n\nHere B is the magnetic field of our Galaxy (B ≈ 5× 10−6G). The spectral index\nβs = −(p + 3)/2 has the value ≈ −2.5. In order to model synchrochon emission\nfor component separation, the variation and uncertainty in the spectral index is\nan important issue in foreground studies. In addition, the galactic cosmic rays\nand the magnetic field are necesary to model accurately and remove foreground\npolarization [34, 38, 146]. Figure 3.15 represents Planck 2018 result of synchrotron\npolarization map at 30 GHz. The larger polarization region at high latitudes are\nthe local structures of the Milky Way. These directions are not good to observe\nCMB polarization.\n\nPComm\ns\n\n10 300µKRJ at 30 GHz\n\nFigure 3.15: The Planck 2018 result of syncrochon polarization at 30 GHz\nusing Commander method with 40′ FWHM resolution [117].\n\n\n\nSystematic effects 76\n\n3.7.3 Free-free\n\nFree-free or thermal Bremsstrahlung is an emission mainly at low frequencies of\nthe electron-ion (warn ionized medium) scattering through the interaction of free\nelectrons with positive charge nuclei in interstellar plasma [34]. It is unpolarized\nemission because of the isotropic and random direction of scattering electrons [60].\nThe spectral index of free-free emission depends only on the electron temperature\nTe. Planck measured the Te ≈ 8000K the the brightness temperature of the free-\nfree emission follows a power law with the spectral index ∼ −2.14 [146].\n\nTB ≈ ν−2.14. (3.66)\n\n3.7.4 Spinning dust\n\nAnomalous microwave emission (AME) or spinning dust is a observed Galactic\nforeground which is compatible with the model of Draine and Lazarian [39] of\nvery fast spinning of nano dust grains. Another candidate for AME is thermal\nfluctuations of the magnetic pole of small silicate grains [146]. The spectral index\nof AME is close to the thermal dust spectral index βAME ≈ −2.5. The AME\nis expected to have a very small amount of polarization and it falls rapidly at\nhigh-frequency [60].\n\n3.8 Systematic effects\n\nIn this section, I will present the most important systematic effects which impact\nto the measurement of CMB polarization from learned lessons of Planck mission\nfor future CMB projects. Figure 3.16 shows the contribution of systematic effect\nto final angular power spectra of Planck high-frequency instrument at 100 GHz\nand 143 GHz [103]. We have instrumental noise, pointing uncertainty, near and far\nsidelobes, ADC nonlinearity, temperature fluctuation. The main systematic effects\nare shown in the table 3.2. Generally, some of the systematic effects produce\nleakage from intensity to polarization signal. These effects lead to observe the\nspurious primordial B mode angular power spectrum. I will introduce some of the\nmain systematics as an example that are relevant for future mission.\n\n\n\nSystematic effects 77\n\nFigure 3.16: Power spectra of Planck HFI systematic effects, the black curves\nare EE angular power spectrum. [108].\n\n\n\nSystematic effects 78\n\nEffect Source Leakage & issue\nCosmic rays at Lagrange point 2 Loss data\n\nGain mismatch Detector pair T → E,B\nBeam asymetry Optical beams T → E, B; E → B\n\n1/f noise RF amplifiers T → E, B\nPointing uncertainty Attitude control, T → E, B\n\nPointing reconstruction,\nBandpass mismatch RF spectral filters T → E, B\n\nHWP Imperfection HWP, 4f noise T → E, B\n. . .\n\nTable 3.2: The main systematic effects table.\n\n3.8.1 Cosmic rays\n\nPlanck had a significant contribution from cismic rays. The cosmic rays are high\nenergy particles: 89 % proton 10 % α particles 1 % nuclei heavier elements and\nelectrons. The main source of CRs at L2 is from the Milky Way galaxy and is\naffected by Solar flares. The flux of cosmic rays at L2 is about 5 cm−2s−1. Figure\n3.17 shows contributions of glitches to the data. Cosmic rays penetrated the focal\nplane and produced glitches in Planck data. The Planck data experiences that\nCRs hit the silicon die, the absorber and the thermometer of bolometers. Their\nthermal energy produces short glitches (the time constant is of the order of a\nfew miliseconds), long glitches (time constants are of the order of 60 ms and 2 s)\nand longer glitches (0.5-5s) and the rate is 2 glitches per second [100, 145]. After\nPlanck, the technology of CMB projects evolved the use of Transition Edge Sensor\n(TES) or Kinetic inductance detector(KID). In order to study behavior respond\ntimes of detectors towards CRs, in the laboratory we can use a radioactive source.\nThe detail of TES time constants study is described in chapter 5.\n\n3.8.2 Beams\n\nThe beam is an instrumental angular response to the signal. We can clasify in\n[107, 120]:\n\n• The optical beam is the response of an optical system which can be horns,\nantenna coupled to the telescope mirror. The optical beam represents the\npure transfer function of the optical system.\n\n\n\nSystematic effects 79\n\nFigure 3.17: The Planck data of some detectors in around 200 seconds, the\ndata is dominated by CMB dipole, galactic and cosmic rays.\n\n• The scanning beam is defined by the beam in-flight. The scanning beam has\nthree different components as the main beam which has angular up to 30’,\nthe near sidelobes have angular between 30’ and 5◦, the far sidelobes have\nangular beyond 5◦.\n\n• The effective beam is defined in the map domain for a given pixel by aver-\naging scanning beams associated with a scanning strategy.\n\nThe convolution of the observed CMB signal and instrumental beam along the\nscanning strategy would produce leakage from intensity to polarization E mode,\nB modes. This convolution also induced mixing of E and B modes due to asym-\nmetric beam so-called elliptical, gain mismatch and pointing mismatch. Beam\nellipticity or beam mismatch is the result of astigmatic aberrations and detector\ntime constants. The gain mismatch is caused by bandpass mismatch of detectors\nleading to different beam response functions. Pointing mismatch is the mismatch\nof center beams. Basically, we can write the signal Si at the measurement i [49]\n\nSi =\n\nˆ\ndΩ bi (Ω) Θ(Ω) + ni. (3.67)\n\nwhere Θ(Ω) is the CMB signal at a solid angle location Ω. ni is the instru-\nmental noise, bi (Ω) is the beam response. There have two approachs to study\nasymmetry of beam reponse systematic effect and correct from the data, the first\nimplementation is directly in real space convolution (Fourier space). The second\n\n\n\nState of the art 80\n\nimplementation is in the spherical harmonics space convolution (angular power\nspectrum).\n\n3.8.3 Bandpass mismatch\n\nThis effect is induced by differences in detector filters. These differences at the\nedges and shape of bandpass filters are measured and described in Planck publi-\ncations [99, 103]. This effect is observed at percent level of variation. Figure 3.18\nillustrates transmitted filters of many detectors of Planck satellite. The bandpass\nmismatch error produces leakage from temperature to polarization. The Bandpass\nmismatch error is an important systematic effect and impacts the estimation of\nprimordial B-mode for future CMB experiments. The bandpass mismatch system-\natic effect is one of my PhD topics, this study is described in detail in Chapter 4.\n\n3.9 State of the art\n\nAs we know the CMB power spectrum is close to a perfect blackbody. Since the\ndiscovery of CMB in 1964 by Penzias and Wilson using horn antenna, there have\nbeen 4 satellites RELIKT-13, COBE(Time observation: 1989-1993), WMAP(Time\nobservation: 2001-2010), Planck(Time observation: 2009-2013) and many ground-\nbased as well as balloon borne experiments. The European Space Agency’s Planck\nmission made public final results in 2018. Figure 3.19 presents results of Planck\n2018 and current different experiments. The temperature TT and E-mode polar-\nization are well fit with concordance 6 parameters ΛCDM model while the next\ngeneration of experiments expects to measure B-mode polarization and will im-\nprove lensing measurements.\n\nAfter two decades of Planck, the temperature anisotropies are measured with high\naccuracy. Now, scientists are moving to smaller projects including ground-based,\nballoon-borne experiments alongside with the space telescopes. The main focus\nis to measure the CMB polarization signal meeting the inflation. There always\n3RELIKT-1 is a Soviet CMB anisotropies experiment. It launched 1 July 1983, its result reported\na blackbody spectrum and anisotropies of CMB in January 1992. Nevertheless, the Nobel Prize\nin physics for 2006 was awarded to COBE team who announced the result on April 23, 1992.\n\n\n\nState of the art 81\n\nFigure 3.18: The spectral filters of Planck satellite. There are variations on\nthe edges and top of filters. (Bottom right) The calibration from dust for each\n\ndetector, blue points are ground test, red points are flight data [99, 103].\n\nhave advantages and disadvantages of satellites, ground-based, balloon-borne ex-\nperiments. A satellite can cover full sky observation while the balloon-borne and\nground-based can only cover a fraction of the sky. Due to constraints of weight\nand cost, the focal plane or the telescope size of satellites and balloons are usually\nlimited in resolution compared to ground-based experiments that use a large tele-\nscope. The obvious difference between satellites and ground-based experiments is\nthe atmosphere of the Earth. The ground-based experiments usually have to op-\nerate at a dry location and high attitude such as the South Pole, Atacama desert\nof Chile and Tibet. The balloon-borne experiments fly at around tens kilometers\nat attitudes with less effect of atmosphere but still receive some emission. Last\nbut not least the time observation for balloons are usually short few days while for\nsatellite and ground are several years [149]. All the observation has to face with\nastrophysical foreground emissions and systematic effects of the experiment.\n\n\n\nState of the art 82\n\nFigure 3.19: The state of the art after Planck 2018 result. The top panel shows\nthe angular power spectrum of temperature TT, and polarization EE, BB. The\nmiddle panel shows the cross-correlation spectrum TE. The bottom panel shows\nthe lensing defection angular power spectrum. Different colours mean different\n\nprojects.\n\n\n\nGround base experiment: QUBIC 83\n\nCurrently, most of CMB projects are on-going or planned by the USA com-\nmunity such as BICEP3/KECK, CLASS, SPT3G, AdvACT, Simon Observatory\n. . . QUBIC, QUIJOTE are ground-based and LSPE is the balloon-borne CMB B-\nmode experiment of the European community. QUBIC experiment is undergoing\nthe construction phase and is preparing to move to Argentina for observing the\nsky at the beginning of 2019. In space, LiteBIRD (Light satellite for the studies of\nB-mode polarization and Inflation from cosmic background Radiation Detection)\nis a proposed satellite to Japan Aerospace Exploration Agency (JAXA). LiteBIRD\ncurrently is in phase A1, concept design study.\n\n3.10 QUBIC and LiteBIRD\n\n3.10.1 Ground base experiment: QUBIC\n\nQUBIC (the Q&U Bolometric Interferometer for Cosmology) is a ground based\nexperiment designed to measure the B-mode polarization signal at intermediate\nangular scales (30 ≤ ` ≤ 200). The science objective is measurement tensor-to-\nscalar r with constraint σ(r) = 0.01 with foreground (0.006 no foreground) with\n95 % confident level as shown in figure 3.20 [31]. QUBIC will observe the sky with\nthree frequencies 90 GHz, 150 GHz and 220 GHz at Alto Chrorrillos in Argentina.\nQUBIC uses a novel kind of instrument, a bolometric interferometer concept. This\ntechnology allows to take advance of interferometry (control systematic effects)\nand bolometer detectors imagers (image the sky on a focal plane and directly\nmeasure temperature in a given direction [31]) which have high signal sensitivity.\nThe table 3.3 describes basic general information as well as requirements of the\nQUBIC experiment from the cryostat, the instruments, the focal plane to scanning\nstrategy.\n\n3.10.1.1 General principle\n\nThe QUBIC bolometric interferometer principle is that a dual reflector telescope\nis selected by horns (diffractive apertures make spatial filtering i.e. the entrance\npupil is a square array of gaussian-illuminated apertures) and then recombined\nbeam. QUBIC uses the Fizeau interferometer approach which is a linear combina-\ntion of summing outputs beam. The correlation between two receivers (detectors)\n\n\n\nGround base experiment: QUBIC 84\n\nFigure 3.20: QUBIC self-calibration sensitivity.\n\ncontains interferometer terms which allow us to access directly the Fourier modes\n(so-called visibilities) of Stokes parameters I, Q and U [31, 84]. We can model\nthe QUBIC instrument using the formalism of Jones matrices which is 2 × 2 di-\nmension. A Jones matrix represents for an instrument, for several instruments the\nJones matrix is simply multiplications matrices. Basically, we can assume that the\nincident radiation has orthogonal electric field amplitude Ex, Ey passing through\nreceivers: (i) A rotating half wave plate before (ii) the polarizing grid and (iii)\nhorns as described in figure 3.21.\n\n(\nErec\nx\n\nErec\ny\n\n)\n= JQUBIC\n\n(\nEx\n\nEy\n\n)\n, (3.68)\n\nhere the Jones matrix for QUBIC instrument\n\nJQUBIC = Jhorn;iJpJ\nT\n\nrotJhwpJrot;\n\n= Jhorn;iJp\n\n(\ncos (ψhwpt) − sin (ψhwpt)\n\nsin (ψhwpt) cos (ψhwpt)\n\n)(\n1 0\n\n0 −1\n\n)(\ncos (ψhwpt) sin (ψhwpt)\n\n− sin (ψhwpt) cos (ψhwpt)\n\n)\n,\n\n\n\nGround base experiment: QUBIC 85\n\nName tag Information\nInstrument Diameter < 1.6 m\nInstrument Height < 1.8m\nInstrument Weight < 800 kg\nWindow diameter 39.9 cm\nFilters diameters 39.2 cm\nPolarizer diameter 32.6 cm\n\nHalf-Wave plate diameter 32.7 cm\nBack-to-back Horn array 400 (diameter 33.078 cm)\n\nOptical combiner focal length 30 cm\nM1 shape and diameter 480mm × 600 mm\nM2 shape and diameter 600 mm × 500 mm\nFrequency channels 150 GHz and 220GHz\n\nBandwidth 25 %\nPrimary beam FWHM at 150 GHz, 220 GHz 12.9◦ , 15◦\nBlue center peak FWHM 150GHz, 220GHz 23.5 arcmin, 16 arcmin\n\nNumber of bolometers / focal plane 1024\nDetector stage temperature goal 320 mK\n\nBolometers NEP 5× 10−17W.Hz−1/2\n\nScientific Data sampling rate 100 Hz\nBolometers time constant < 10 ms\n\nTES size 2.6 mm\nRotation in azimuth -220◦ / + 220◦\nRotation in elevation +30◦ / +70◦\n\nRotation around the optical axis -30◦ / +30◦\nPointing accuracy < 20 arcsec\nAngular speed Adjustable between 0 and 5◦/s\n\nwith steps < 0.2◦/s\n\nTable 3.3: QUBIC experiment general information [31].\n\n= Jhorn;iJp\n\n(\ncos (2ψhwpt) sin (2ψhwpt)\n\nsin (2ψhwpt) − cos (2ψhwpt)\n\n)\n. (3.69)\n\nHere Jhorn;i is the Jones matrix for the horn i. Jp is the Jones matris for the\npolarizing grid. ψhwp is the angular velocity of the half-wave plate. Jrot and\nJhwp are the rotational matrix and the ideal Jones matrix of the half wave plate,\nrespectively. After the QUBIC’s receiver system, an ideal detector j measures the\nsignal at time t, for the frequency ν is [84]\n\nS (j, ν, t) = SI (j, ν) + SQ (j, ν) cos (4ψhwpt) + SU (j, ν) sin (4ψhwpt) . (3.70)\n\n\n\nGround base experiment: QUBIC 86\n\nFigure 3.21: The basic elements of the QUBIC’s cryostat are cooled down to\n320 mK at the focal plane, the optical system is worked at the level of 1 K.\nThe quasi-optical system including a half-wave plate, polarizing grid and horns\n\nsystem is maintained at 4 K.\n\nThese terms SI,Q,U are intensity and polarization signal convolved with the syn-\nthetic beam. Due to the fact that the half-wave plate and horns have imperfection.\nTherefore, in order to study systemmatic errors, we introduce the complex gain\nparameters gx, gy, hx, hy in diagonal terms and the complex coupling parameters\nex, ey, ξx, ξy in non-diagonal terms [21] [88], finally we have the Jones matrix for\nthe half-wave plate is\n\nJhwp =\n\n(\n1− hx ξx\n\nξy −(1 + hy)\n\n)\n. (3.71)\n\nThe Jones matrix for the horn i\n\nJhorn;i =\n\n(\n1− gx;i ex;i\n\ney;i 1− gy;i\n\n)\n. (3.72)\n\nThe nature of the bolometric interferometer of QUBIC is so-called self-calibration.\n\n\n\nGround base experiment: QUBIC 87\n\nWindow, HWP, polarizing grid\n\nHWP rotator\n\nHorn array\n\nSwitches\n\nPulse tubes\n\nMirrors\n\nDichroic\n\nFocal plane\n\nFigure 3.22: QUBIC instrument\n\nThe basic idea is that the image of an open horn pair (named a baseline) is\nrepeatably observed with many different pairs (redundant baselines). So that\nthe systematic errors are able to control. The mechanism of horns is that they\nhave shutter switches which are placed between primary and secondary horns.\nThe switches can on/off a single horn pair. This procedure allows measuring the\nimage of a polarized source with all baselines or a fraction of baselines and control\ninstrumental systematic effects.\n\n3.10.1.2 Instrument\n\nThe QUBIC instrument is located inside a cryostat which is cooled down to 4K\nusing pulse-tubes. Figure 3.21 and 3.22 shows QUBIC’s instrument in its cryostat.\nThe quasi optical components (mesh filters, HWP, polarizer, dichroic) are man-\nufactured by the Astronomy Instrumentation Group (AIG) in Cardiff with high\nTRL technology readliness level [31]. An open window (high density polyethy-\nlene) ∼ 45 cm scan the incident sky light with set of filters. The rotating half\n\n\n\nSpace satellite mission: LiteBIRD 88\n\nwave plate and Polarizing grid: The QUBIC’s HWP made of metamateri-\nals with the embedded mesh filters technology. The bandwidth requirement of 2\nchannel frequencies is 73 %. The rotational QUBIC’s HWP (3K) modulates the\npolarization using a rotational mechanism and a stepping rotator (300K) mounted\noutside the cryostat. The rotational HWP has 8 positions corresponding to 11.25◦\n\nfor a step. Polarization states are selected by a polarizing grid [31]. Horns,\n\nswitches: The feed horn array contains 400 back-to-back horns with movable\nswitches in the middle. The shutters of switches can open or close independently\nthe optical path. This mechanism is used in the self-calibration phase. Mirrors:\n\nThere have 2 Aluminium mirrors for the optical combiner. Each of them has 9\nattached supported points which allow to alignment the mirror system. Focal\nlengths of the primary and the secondary mirror are 231 mm, 196 mm respec-\ntively. They are set at a distance of 578 mm. Thus the focal length of the system\nis ≈ 300 mm. Dichroic/polarizer/filters is an ecliptic optical element which\nis designed to transmit the 220 GHz band and reflect the 150 GHz band with\nan efficiency of > 90%. The focal plane of QUBIC will be used around 2048\nTransition Edge Sensors (TES) detectors with total noise equivalent power NEP\n∼ 5× 10−17WHz−1/2. Each frequency band has 4 arrays of 256 TES pixels. The\nrequirement of time constants is in the range of 10-100 ms.\n\n3.10.2 Space satellite mission: LiteBIRD\n\nThe concept design of the LiteBIRD spacecraft is shown in figure 3.23. The pay-\nload module consists of the low-frequency telescope and the high-frequency tele-\nscope with half-wave plate, the focal plane, cryogenic system. The Service module\nof the BUS module supports, power supply, communication system. The mass and\nconsumption power of LiteBIRD is estimated at 2.6 tons and 3.0 kW, respectively\n[73]\n\nThe science goal of LiteBIRD is to detect B-mode polarization and to measure\ntensor-to-scalar r of the order of 10−3. The concept design of LiteBIRD is an\noptimized mission of light and small satellite for the B-mode detection at large\nscale to intermediate scale (2 ≤ ` ≤ 200). The key concept of LiteBIRD is the\nhalf-wave plate (HWP) modulation. The satellite will be located at the Sun-Earth\nLagrange point 2 with 3 years observation. The study of an optimized scanning\nstrategy of LiteBIRD is described in the chapter 4, the scanning parameters are\n\n\n\nSpace satellite mission: LiteBIRD 89\n\nLFT (5K)\n\nHG-antenna\n\nHFT (5K)\n\nV-groove\n\nradiators\nSVM/BUS\n\nPLM\n\n200K\n100K\n30K\n\nJAXA\nH3\n\n4.5 m\n\nFigure 3.23: The concept design of the LiteBIRD spacecraft which composed\nof the payload module (PLM) and the service module (SVM).\n\nprecession angle α, spin angle β, precession period τprec and spin period τspin as\nshown in figure 4.1. The instrument is designed to sense ∼ 4.1 µK. The Low-\nFrequency Telescope (LFT) 40 GHz - 235 GHz has a 400 mm aperture, Crossed-\nDragone telescope. The transition edge sensor (TES) detectors array of LFT has\nbeen developed for the POLARBEAR experiment by the University of California\n(UC) Berkeley and UC San Diego. The High-Frequency Telescope (HFT) from\n280 GHz to 400 GHz has a 200 mm aperture refractor with two silicon lenses.\nThe TES array of HFT with corrugated feedhorn has been developed for ABS,\nACTpol, SPTpol by UC Boulder, NIST, and Stanford. The layout of the focal\nplane unit and the main cold system are shown in figure 3.24.\n\nFigure 3.24: The concept design of the cold system needs to be cooled to ∼\n5K. The focal plane unit of the low, mid-frequency, the dimension of the unit is\n\n420 mm × 600 mm.\n\nHalf wave plate\n\nLiteBIRD will use continuous rotating half-wave plates (HWP) for both telescopes.\nThe HWP can modulate the polarized angle of an incident radiation, mitigate sys-\ntematic effects as 1/f noise by shifting the signal above free frequency, bandpass\n\n\n\nSpace satellite mission: LiteBIRD 90\n\nmismatch as well as beam mismatch [142]. The polarization signals can be mea-\nsured by a single detector, thus it removes out systematic error of combining\nmulti-detector. Basically, HWP (or birefringence) is an optical element called as a\nretarder. The output from the HWP will have different phases and different polar-\nizations. The incident angle will be rotated twice after an HWP. The characteristic\nequation is the retarder of a single wave plate [80]\n\n∆δ = 2π\nν\n\nc\n|n0 − ne|d = (2m+ 1)\n\nλ0\n\n2\n. (3.73)\n\nhere ν is the frequency and λ0 is its equivalent wavelength, d is the thickness of\nthe material, n0, ne are the ordinary and extraordinary refractive indices of a wave\nplate. m=0,1,2,3. . . is the number of wave plates in a stack HWP. We can study\ntransformation of radiation through a HWP using Mueller 4× 4 matrix formalism\nwhich is an overlapping generalization of 2× 2 Jones matrices:\n\nM = A (J⊗ J∗) A−1, (3.74)\n\nwhere ⊗ is the tensor product, ∗ is the complex conjugate and\n\nA =\n\n\n\n\n1 0 0 1\n\n1 0 0 −1\n\n0 1 1 0\n\n0 −i i 0\n\n\n\n. (3.75)\n\nFully polarized light can be treated by Jones formalism or Mueller formalism while\nunpolarized or partially polarized light must be treated by Muller formalism. In\nthe literature both formalism are used for HWP. We described Jones formalism\nin the section 3.10.1, this is a motivation to use the two formalism in this thesis.\nThe output signal along the z axis with rotation angle ρ of the HWP around its\naxis after m wave plates is [80]\n\nSout =\nm∏\n\ni=1\n\nR (−ρ) M R (ρ)\n\n\n\n\nI\n\nQ cos 2ψ\n\nU sin 2ψ\n\nV\n\n\n\n. (3.76)\n\n\n\nSpace satellite mission: LiteBIRD 91\n\nψ is the polarized angle of incident radiations. An ideal HWP has the Muller\nmatrix:\n\nM =\n\n\n\n\n1 0 0 0\n\n0 1 0 0\n\n0 0 −1 0\n\n0 0 0 −1\n\n\n\n, (3.77)\n\nwith a reference frame along the ordinary and extroordinary axis. The rotation\nmatrix is introduced as\n\nR (ρ) =\n\n\n\n\n1 0 0 0\n\n0 cos 2ρ sin 2ρ 0\n\n0 − sin 2ρ cos 2ρ 0\n\n0 0 0 1\n\n\n\n. (3.78)\n\nDue to the fact the the HWP is imperfection, in order to study the systematic of\na HWP, the Muller matrix can be expressed as\n\nM =\n\n\n\n\nMII MIQ MIU MIV\n\nMQI MQQ MQU MQV\n\nMUI MUQ MUU MUV\n\nMV I MV Q MV U MV V\n\n\n\n, (3.79)\n\nand those coefficients in the matrix can be obtained by simulation tools or exper-\nimentation in laboratory. The equation 3.76 is useful for simulations and study\nLiteBIRD measurements as map making, and systematic effects.\n\nThe focal plane of LiteBIRD\n\nLiteBIRD will observe the sky with 15 frequency bands from 40 to 400 GHz of\nabout 2622 TES detectors [78, 79]. The frequency bands have been defined to\navoid CO lines. The observing frequency bands, bandwidth, noise equivalent power\n(NEP) are showed in the table 3.4 and figure 3.25. The focal plane unit is cooled\ndown to ∼ 100 mK. The fabrication of the LiteBIRD focal plane unit will be\nimplemented at the NIST Microfabrication facility in Boulder and the Marvell\nNanofabrication laboratory in Berkeley. The low/mid frequency array technology\nis multi-chroic lenslet coupled sinuous antenna detectors. This technology has been\nimplemented for POLARBEAR, Simons Array, SPT-3G and Simon Observatory\nexperiments. The low frequency has a baseline of 18 mm diameter per pixel.\n\n\n\nSpace satellite mission: LiteBIRD 92\n\nType ν BW Beam NPix Loading NETCMB,Arr Sensitivity\n[GHz] [%] [arcmin] [pW] [µK · √s] [µK− arcmin]\n\nLF-1 40 30 69.2 114 0.15 25.0 36.8\nLF-2 50 30 56.9 114 0.18 16.0 23.6\nLF-3 60 23 49.0 114 0.16 13.2 19.5\nLF-4 68 23 40.8 114 0.17 10.8 15.9\nLF-5 78 23 36.1 114 0.19 9.0 13.3\nLF-6 89 23 32.3 114 0.20 7.8 11.5\nMF-1 100 23 37.0 296 0.18 6.1 9.0\nMF-2 119 30 31.6 222 0.25 5.1 7.5\nMF-3 140 30 31.6 296 0.25 3.9 5.8\nMF-4 166 30 24.2 222 0.24 4.3 6.3\nMF-5 195 30 21.7 296 0.22 3.9 5.7\nMF-6 235 30 19.6 222 0.18 5.1 7.5\nHF-1 280 30 13.2 128 0.13 8.8 13.0\nHF-2 337 30 11.2 128 0.10 13.0 19.1\nHF-3 402 23 9.7 128 0.05 25.0 36.9\n\nTable 3.4: Summary of detector configuration and sensitivity, LF, MF and HF\nstand for low, mid and high frequency respectively. BW is fraction of bandwidth.\nNET stands for noise equivalent temperature as well as noise equivalent power.\n\nC\nO\nJ1\n\n0\n\nC\nO\nJ2\n\n1\n\nC\nO\nJ3\n\n2\n\nC\nO\nJ4\n\n3\n\ndu\nst\n\nsynchrotron\n\nCMB\n\nLFT\nHFT\n\nFigure 3.25: LiteBIRD frequencies.\n\n\n\nSpace satellite mission: LiteBIRD 93\n\nFigure 3.26: The low/mid frequency technology: Top: The low, mid frequency\npixel is fabricated by the Marvell Nanofabrication laboratory in Berkeley [152].\nBottom: The main components of a pixel are the sinuous antenna at the center,\nfour diplexer bandpass filters, four TES surround these filters. A pixel has 3\n\nmillimeters in diameter.\n\nThe mid frequency has a baseline of 12 mm diameter. The bandwidth, number\nof pixels, beam size are given in the table 3.4. Each pixel is a dual polarized\nsinuous antenna (log-periodic) coupled with radio frequency transmission lines,\nChebyshev bandpass filters 4 for several frequency bands and TES bolometers\nwhich measure power in each frequency band. The sinuous antenna-coupled is\na four-armed antenna with the self-similar structure (16-cell) as shown in figure\n3.26. The sinuous antenna is (i) sensitive to the CMB linear polarization by\na pair of opposite arms if we have power and 180◦ phase difference, (ii) planar\nantenna for large arrays, (iii) the sinous antena has the inner and outer radius\nof antenna defining the frequency bands (broadband), (iiii) high gain amplitude\n4Comparing Butterworth and Chbyshev filters, we know that a Chebyshev filter has a sharp rise\nand steeper drop thus the cut off frequency is better determination.\n\n\n\nSpace satellite mission: LiteBIRD 94\n\nwhich is compatible with telescope, and a stable impedance. The sinuous coupled\nantenna is placed under lenslet in order to boost the gain of an antenna [40, 141].\n\n337\t\r  GHz\t\r  \n\n280\t\r  GHz\t\r  \n\n402\t\r  GHz\t\r  \n\nb)\t\r  \n\nc)\t\r  \nd)\t\r  \n\ne)\t\r  \n\n150\t\r  mm\t\r  diameter\t\r  wafer\t\r  \n\na)\t\r  \n\nbackshort\t\r  \t\r  \ncap\t\r  \n\nbackshort\t\r  \n\ndetector\t\r  \n\ninterface\t\r  \nfeedhorn\t\r  \n\nFigure 3.27: The high frequency technology: a) The high frequency pixel is\nfabricated by NIST. The technology shows the feedhorn coupled array and the\nreadout board. b) A detector stack assembly. c) The hexagonal detector array\nhas three frequency bands in a 150 mm diameter wafer. Each band has 64 pixels\nwith only one low pass filter. d) Zoom in at the center of the array. e) Zoom in\n\na single pixel design which indicates two orthogonal polarized bolometers.\n\nThe high-frequency array technology is the single color of an orthomode transducer\n(OMT) coupled corrugated horn detectors. This technology has been implemented\nfor ACT-pol, SPT-pol and SPIDER experiments as shown in these figures 3.27.\nEach frequency band has 64 dual polarized pixels. The orthomode transducer\ndetector technology has high Technology readiness level (TRL) level at high fre-\nquency, symmetry of beam, clean polarized property [142]. The corrugated feed-\nhorns, a gold-plated silicon-platelet, transmit radiation through a coplanar waveg-\nuide (CPW) directly to microstrip (MS), then isolated TES bolometers area [87].\nThe OMT circle design has the orthogonal polarization independently for each\n\n\n\nBandpass Mismatch 95\n\nfrequency, the incident radiation will be transmitted to MS by CPW transmission\nlines independently from OMT then the signals pass through the bandpass filter\nwhich is set of resonant MS in a pixel. Now the incident signals are measured for\neach polarization by TES [43, 59, 81, 82].\n\n\n\nChapter 4\n\nBand-pass mismatch\n\nContents\n4.1 Sky emission model and mismatch errors . . . . . . . . 98\n\n4.2 Calculating the bandpass mismatch . . . . . . . . . . . 104\n\n4.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 107\n\n4.2.2 Analytic estimates . . . . . . . . . . . . . . . . . . . . . 119\n\n4.2.3 Precession period and spin period ratio τprec/τspin . . . . 125\n\n4.3 A correction method . . . . . . . . . . . . . . . . . . . . 133\n\n4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 136\n\nThe future Cosmic Microwave Background (CMB) satellite concepts LiteBird [79],\nCORE [32], PIXIE [67] have been proposed to probe B modes polarization to\nmeasure the tensor-to-scalar r ratio with a sensitivity σ(r) ≤ 10−3, which is almost\ntwo orders of magnitude beyond the Planck sensitivity. LiteBIRD is a proposed\nCMB polarization satellite project to JAXA aims to probe the inflationary B-mode\nsignal [45, 62, 78, 79]. It will observe the full sky with more than 2000 detectors,\nfrequency coverage spans 40–402 GHz from the second Lagrange point for 3 years.\nLiteBIRD is an optimized satellite for the observation of the large-scale of the CMB\nB-mode polarization and it is a reasonably light and small satellite to maximize\nthe chance for launch. LiteBIRD has a half-wave-plate modulation, which rotates\nthe polarization of linearly polarized light. Typical CMB experiments observe in\na number of different frequency channels with many detectors for each channel.\nThe effect of non-uniformity, or unbalance, of the bandpass filters for different\ndetectors induces leakage from intensity to polarization after calibrating the data\n\n96\n\n\n\nBand-pass mismatch 97\n\non the CMB. In this chapter, I will present the results of the study of the bandpass\nmismatch systematic effect.\n\nAn#-­‐Sun\t\r  direc#on\t\r  \n\nSpin\t\r  angle\t\r  β \t\n\r\n\nPrecession\t\r  \nangle\t\r  α\t\n\r\n\nτspin\t\r  \n\nτprec\t\r  \n\nFigure 4.1: Representation of typical satellite scanning strategy, the spin angle:\nβ, precession angle: α, rotating spin: τspin or ωspin, precession spin: τprec or ωprec.\n\nMy study first focuses on the evaluation of the level of the bandpass mismatch\neffect impact on the final determination of the tensor-to-scalar ratio r in the case\nwithout a Half Wave Plate (HWP) at 140 GHz. I have studied also bandpass\nmismatch effect for CORE scanning strategy parameters. I have done a code for\nbandpass mismatch simulation based on a co-addition map making method. The\ncode has been integrating into the Japanese LiteBIRD simulation tool package,\nwhich operated in the high energy accelerator research organization known as KEK\ncomputers. The final tool intergrating previous work can simulate the boresight\npointing, the data and the map making for the full focal plane of LiteBIRD. As\na conclusion of this study, the level of leakage depend strongly on the scanning\nstrategy of a satellite (α: Precession angle and β: Spin angle) which are illustrated\nin figure 4.1. Furthermore, we have studied the importance of choosing scanning\nstrategy parameters ratios ωprec\n\nωspin\nto avoid Moiré patterns which produce features on\n\nthe power spectrum. The results of the study allow us to choose the scanning strat-\negy of future CMB satellites. In addition, I have verified the analytic estimation\n\n\n\nSky emission model and mismatch errors 98\n\nof the bandpass mismatch effect with simulation. The method indicated the tight\ncorrelation between leakage maps and crossing moment maps 〈cos 2ψ〉 , 〈sin 2ψ〉,\nthe analytic estimation method is a useful approach to cross-check, fast and easy\nways to predict the magnitude of potential leakage. Moreover, I also studied a\ncorrection method for the bandpass mismatch error systematic effect.\n\nThe amplitude of the angular power spectrum is decreased when the precession\nangle α is larger, as explained by a more uniform angle coverage for larger α. The\nmagnitude of the tensor-to-scalar in the reionization bump (2 ≤ ` ≤ 10) is of the\norder of 10−3, in the recombination bump (10 ≤ ` ≤ 200) is of the order of 10−5\n\ndepending on scanning angle parameters. So that the bandpass mismatch error\nis not a negligible effect at the reionization bump that can impact to current and\nnext generation of CMB polarization missions.\n\nIn addition, I also verified that the bandpass mismatch effect is greatly mitigated\nwith an ideal rotating HWP without any achromaticity or other non-idealities.\nBut the imperfect HWP has systematic effects.\n\nThe simulation pipeline is illustrated in the flowchart figure 4.2. These processes\nstart from the filter coefficient to the sky simulation, the map making equation\nand the computation of angular power spectra. For more details, this systematic\neffect has been discussed in the paper \"Bandpass mismatch error for satellite\nCMB experiments I: Estimating the spurious signal\", Journal of Cosmology and\nAstroparticle Physics December 2017 [50].\n\n4.1 Sky emission model and mismatch errors\n\nIn order to model the emission of the sky, we only use CMB intensity and thermal\ndust intensity as a start. The input maps at 140GHz are shown in figure 4.3 for\nCMB intensity and thermal dust intensity in µK.\n\nFor the integration on the frequency band, we need to model the spectra of com-\nponents. As we known, the spectrum of CMB is a black body at T=2.7255K. We\nremind the Planck function with respect to temperature and frequency:\n\nB(ν; T) =\n2hν3\n\nc2\n\n1\n\nexp( hν\nkT\n\n)− 1\n[Wsr−1m−2Hz−1]. (4.1)\n\n\n\nSky emission model and mismatch errors 99\n\nAngular Power SpectrumAngular Power Spectrum\n\nData SimulationData Simulation\nd= ICMB+γd I dust\n\n(AT N−1 A)−1 AT N−1d\n\nMask 20%Mask 20%\n\nC l\nBB ,C l\n\nEE\n\nFilter SimulationFilter Simulation\nγd\n\nScanning \nstrategy\n\nScanning \nstrategy\n\nDust mapDust mapCMB mapCMB map\n\nFigure 4.2: The simulation process\n\nT is the brightness temperature of the source. We often express the sky emission in\nCMB temperature units. The relationship between intensity I and δT is obtained\nby linearizing the Planck formula around T = T0:\n\nδI(ν)\n\nδT\n=\n\n(\n∂B(ν; T)\n\n∂T\n\n)\n∣∣T0\n\n=\n2h2ν4\n\nkc2T2\n0\n\nexp( hν\nkT0\n\n)\n(\n\nexp( hν\nkT0\n\n)− 1\n)2 . (4.2)\n\nThe spectrum of thermal dust is a modified blackbody. In the sub-mm domain,\n\nthe dust is optically thin Idust(p̂, ν) = τ(p̂,ν0)\n\n(\nν\nν0\n\n)βd(p̂)\n\nB (ν; Td) where τ(p̂,ν0) is the\noptical dept at frequency center ν0, βd(p̂) is the spectral emissivity index in a\nposition, its value is in range 1.5 − 2. Assuming that the spectral index is a\n\n\n\nSky emission model and mismatch errors 100\n\nFigure 4.3: Input intensity I maps containing CMB and thermal dust at\n140GHz in galactic (left) and ecliptic (right) coordinates. The unit is in µK\n\nconstant, we can relate the intensity at two different frequencies with the following\nformula for any location on the sky:\n\nIdust(p̂, ν) =\n\n(\nν\n\nν0\n\n)βd B(ν; Td)\n\nB(ν0; Td)\nIdust(p̂, ν0). (4.3)\n\nWe assume the same emission law for intensity and polarization, and a constant\nspectral index over the sky but this is not critical for this study.\n\nGenerally, the total intensity Itot(p̂, ν) of the microwave sky can be expressed as\nthe sum of components, the model of the unpolarized sky is:\n\nItot(p̂, ν) = I0(ν) +\n∂B(ν;T )\n\n∂T\n\n∣∣∣∣∣\nT0\n\n∆TCMB(p̂) +\n∑\n\n(c)\n\nI(c)(p̂, ν) (4.4)\n\nwhere B(ν;T ) is the spectrum of a blackbody at temperature T , T0 is the average\nCMB temperature of about 2.7255K, ∆TCMB(p̂) is the CMB temperature fluctu-\nation around this mean value, I(c)(p̂, ν) the emission spectrum of component (c)\n\nas a function of electromagnetic frequency ν, I0(ν) is the monopole including all\ncomponents. The carbon monoxide (CO) emission at transition line frequencies\nν = 115GHz for J = 1→ 0 . . . was a main source of leakage in Planck experiment\n[97], the future satellite will avoid these lines. We have similar relationships for\nthe Q and U Stokes parameters.\n\nLet us define gi(ν) is the bandpass filter transmission for detector i. The intensity\nI(p̂, ν) and polarization Q(p̂, ν) and U(p̂, ν) are the result of the integration of the\n\n\n\nSky emission model and mismatch errors 101\n\nemission of components on the detector band-pass.\n\nI(p̂, ν) =\n\n´\ngi(ν)Icmb(ν)dν +\n\n´\ngi(ν)Idust(ν)dν + ...´\n\ngi(ν)dν\n, (4.5)\n\nwhere Icmb(ν) and Idust(ν) are the intensity of cmb and dust at the frequency ν.\nThe denominator term\n\n´\ngi(ν)dν is for the normalization. We then inject equations\n\n4.3 and 4.2 into the equation 4.4. We obtain the fluctuation of the signal measured\nby the detector i:\n\nˆ\ndν gi(ν)\n\n(\nI(p̂, ν)− I0(ν)\n\n)\n=\n\nˆ\ndν gi(ν)\n\n∂B(ν; T)\n\n∂T\n\n∣∣∣\nT0\n\n∆TCMB(p̂)\n\n+\n\nˆ\ndν gi(ν) Idust(p̂, ν0)\n\n(\nν\n\nν0\n\n)βd(p̂)\nB(ν; Td)\n\nB(ν0; Td)\n+ . . . , (4.6)\n\nwhere ν0 is the central frequency of a band. I0(ν) = B(ν; T0) is the CMBmonopole.\nIdust(p̂, ν0) is the intensity of the dust component at the reference frequency ν0, and\nwhere these dots stand for other components (such as synchrotron and free-free)\nnot explicitly written here. For our study we assume that the galactic thermal dust\nemission is a greybody of temperature Td ≈ 19.7K [98] with an emissivity spectral\nindex β(p̂), which depends on sky positions and whose average value is ≈ 1.62 as\nmeasured by Planck [98, 102]. The synchrotron and the free-free emissions can\nbe described by power law spectra with the negative spectral indices ≈ −3.1 and\n≈ −2.3, respectively [104].\n\nThe CMB temperature is a constant and independent in frequency observations.\nTherefore, we divide two sides of equation 4.6 for\n\n´\ngi(ν)\n\n(\n∂B(ν;T)\n∂T\n\n)\n|T0\n\ndν and ex-\n\npress to the first order we obtain for the total sky intensity Isky(ν0) after converting\n\nthe CMB temperature ∆TCMB (by multiplying with\n(\n∂B(ν0; T)\n\n∂T\n\n) ∣∣∣\nT0\n\n) to intensity\n\nICMB(ν0):\nIsky(ν0) = ICMB(ν0) + γd Idust(ν0) + γs Isync(ν0) + . . . , (4.7)\n\nwhere\n\nγd =\n\n\n\n\n´\ndν gi(ν)\n\n(\nν\nν0\n\n)βd B(ν;Td)\nB(ν0;Td)´\n\ndνgi(ν)\n(\n∂B(ν;T )\n∂T\n\n) ∣∣∣\nT0\n\n\n\n(\n∂B(ν0;T )\n\n∂T\n\n) ∣∣∣\nT0\n. (4.8)\n\nThe factor γs is similarly defined integrating over the synchrotron spectrum, etc.\nThe equation (4.7) also holds for the polarization when I is replaced with Q and\n\n\n\nSky emission model and mismatch errors 102\n\nU. The unit normalization for the CMB component is justified because the data\nare calibrated using the CMB dipole (or higher order temperature anisotropies).\n\nDifferences in the bandpass function gi(ν) from detector to detector result in corre-\nsponding variations in γ from detector to detector for each non-CMB component.\nSuch variations have been observed in Planck data (see Figs. 5 and 28 of [99]\nfor the measured Planck filters and the mismatch parameters, respectively). The\nvariations of the bandpass functions of the filters from a detector to a detector\nfor a future satellite experiment will depend on the kind of detector technology\nused (see also [63] regarding the WMAP experiment). As already stressed, for\nthe above sky emission model where each component has a fixed (factorizable)\nfrequency dependence, the bandpass mismatch maps depend only on the γ pa-\nrameters and not on the other details of the filters. Consequently, the intensity to\npolarization leakage due to bandpass mismatch can be obtained using only the γ\nand no additional properties of the bandpass functions. The γd parameter in front\nof Idust are close to unity when the bandwidth is narrow bands and differences in\ngi(ν) will induce some variations around 1.\n\n110 120 130 140 150 160 170\nν [GHz] \n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\n1.01146596266\n\n1.01598753979\n\n1.01636848863\n\n1.01692558851\n\n1.01827837074\n\n1.0054427877\n\n1.01167149602\n\n1.01113072184\n\n1.01318314\n\n1.02223412137\n\n110 120 130 140 150 160 170\nν [GHz] \n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\n1.00313256688\n\n0.99903052841\n\n0.995298932723\n\n0.996080525965\n\n0.9990645836\n\n0.999748822593\n\n0.997948715757\n\n0.994467192901\n\n1.01004192761\n\n1.00287157864\n\nFigure 4.4: The simulation of bandpass miss-match function at ν0 = 140GHz\nfor 10 detectors, the left is the boxcar function which is vary on the edge of a\nfilter and the right is the tophat fuction which is vary at the top with the surface\n\n1 GHz.\n\nAt this stage, we use a boxcar function for the bandpass g(ν) = Π(ν−ν0\n∆ν\n\n) (figure\n4.4), for which g(ν) = 1 in the interval [νmin, νmax] and g(ν) = 0 elsewhere. The\ncenter frequency is ν0 = 140 GHz and the equation (4.8) uses to simulation band-\npass mismath γd parameter with CMB temperature T0 = 2.725K, thermal dust\nTd = 19.7K, βd = 1.62, but in principle we could use any function. We assume\nthat the variations in νmin and νmax for each detector are generated independently\n\n\n\nSky emission model and mismatch errors 103\n\naccording to a uniform distribution with a width of 1%1. We also assume a band-\nwidth (νmax−νmin)/ν0 of 0.25 on average, with ν0 = 140.7GHz. We also simulated\nthe tophat function as figure 4.4 on the right with average 1 GHz surface of the\ntophat.\n\nIn some micro-fabricated technologies for Transition Edge Sensor (TES) or Ki-\nnetic Inductance Detector (KID), each detector is designed of a feed antenna in a\nsingle band or broadband, a superconducting radio frequency (RF) filter circuit,\nand a transmission termination line of the characteristic impedance. A frequency\nband is defined by the combination of the antenna impedance and the integrated\nRF filter circuit. The fabrication of a bandpass filter on a silicon wafer with a\nNiobium ground plan, a dielectric insulator, and a Niobium strip layer actually\ncould contribute variously to non-ideality in reality. These fabrication parameters\nare:\n\n• Layer to layer misalignment: Due to the machines during fabrication,\nmisalignment could contribute to the mismatch between two orthogonal de-\ntectors. In practical experience, the fabricated machines can align each layer\nup to 0.5 µm. However, we believe this effect does not shift much the shape\nof the filter.\n\n• Line width: The center frequency of a bandpass filter could shift as a\nfunction of line width this error could introduce 1 GHz per 0.2 micron of\nchanging width due to non-uniform etch a part of a lumped filter.\n\n• Dielectric constant: The center frequency of a bandpass filter depends\non the dielectric constant because of changing capacitance of capacitors. If\nwe change 0.1 in the dielectric constant, the center frequency of a bandpass\nfilter will shift 1 GHz. We expect that the value of the dielectric constant is\nfixed on the whole silicon wafer.\n\n• Dielectric thickness: The impedance of a microstrip line and the capaci-\ntance of parallel capacitors depend on the dielectric thickness of a material.\n\n1We thank Aritoki Suzuki for sharing with us that the measurement errors with Fourier Trans-\nform Spectrometer (FTS) in the bandpass of the third-order Chebyshev filter placed between\nthe broadband sinuous antennas and the bolometers of the focal plane panels of the Simons\nArray [152] give approximately this spread. Obviously, since these are values dominated by mea-\nsurement error, the actual bandpass mismatch for these filters could be much smaller. These\nmeasurements merely serve to establish an upper bound on the mismatch. These values are also\nof the same order of magnitude as the values representing the bandpass mismatch of the metal\nmesh filters used as part of the Planck satellite HFI instrument. [See [103] for a discussion of\nthe Planck bandpass mismatch.]\n\n\n\nCalculating the bandpass mismatch 104\n\nFor example, if we have 100 A (Angstrom unit ∼ 10−10) change in the thick-\nness, the center band of a lumped filter will vary 1GHz.\n\n• Kenetic Inductance: The quality of Niobium could also change due to\nincident radiations. The changing quality effect to the kinetic inductance of\na microstrip filter [65]. If we have a change of 0.03 pH/square in the kinetic\ninductance, we will make 1 GHz shift in the frequency band.\n\nThe resulting RMS of γd simulation is 0.6% for these simple filter models. This\nis similar to the variations observed for Planck at 143GHz. The fact that actual\nbandpass functions are more complex functions of ν does not affect the applica-\nbility of the present work as long as the corresponding γ coefficients remain of\nthe same order of magnitude. Results for other values may be obtained by trivial\nrescaling. We verified the expected linear scaling by increasing the width of the\nuniform distribution from 1% to 2% and observed that the leakage increases by a\nfactor of 2, as expected.\n\n4.2 Calculating the bandpass mismatch\n\nIn this section, we assume a simplified model of the single source of the systematic\neffect. We study the bandpass mismatch error in isolation and in the simplest\npossible context. We assumed that the beams are azimuthally symmetric and\nidentical. We also assumed that the monopole and the dipole, as well as gain\nvariations, are already well calibration.\n\nThe scanning strategy of a satellite is characterized by the following parameters: α\n(precession angular radius), β (spin angular radius), τprec (precession period), and\nτspin (spin period). The motion of a satellite and definitions of the scanning param-\neters are indicated in Figure 4.1. Many of the proposed future CMB polarization\nspace missions have adopted such a scan strategy [35, 79].\n\nThe large focal plane of the LiteBIRD mid-frequency spreads over 10 degrees wide.\nFigure 4.5 indicates the position of pair detectors in the focal plane for 5 wafers.\nEach wafer contains 37 orthogonal detectors. As described in [79], 222 detectors\nare designed at 140 GHz.\n\n\n\nCalculating the bandpass mismatch 105\n\nFigure 4.5: The position of 370 detectors at 140 GHz in the 5 wafers of the\nfocal plane, each position is a pair of orthogonal detectors.\n\nI used HEALPix 2 [46] (with nside = 256) to simulate the celestial sphere pixelized\nmaps. In order to obtain several hits in a pixel, a fast sampling rate is chosen,\nthe sampling rate parameter does not affect to the results of the study of the\neffect comparing with the scanning strategy parameters. We also assumed white\ninstrument noise, and we solve the map making equation as a solution of maximum\nLikelihood:\n\nm̂ = (ATN−1A)−1(ATN−1d). (4.9)\n\nHere m̂ includes the estimated maps of Stokes parameters Î , Q̂ and Û . A is\nthe pointing matrix. N is the noise covariance matrix in the time domain, ψ is\nthe polarization angle of a detector with respect to a reference axis. The data\nmeasurements vector d are given by\n\nSj = I(p) +Q(p) cos 2ψj + U(p) sin 2ψj + nj (4.10)\n\nwhere nj represents a stationary white noise source for observations indexed by\nj. Here the index j (j = 1, . . . , Np) labels the observations falling into the pixel\nlabeled by p. Under the hypothesis of white instrument noise, the map making\n2http://healpix.sourceforge.net\n\n\n\nCalculating the bandpass mismatch 106\n\nequation 4.9 can be expressed into a block diagonal form for different pixels.\n\n\n\n\nÎ(p)\n\nQ̂(p)\n\nÛ(p)\n\n\n\n\n=\n1\n\nNp\n\n×\n\n\n\n\n1 〈cos 2ψj〉 〈sin 2ψj〉\n\n〈cos 2ψj〉\n1 + 〈cos 4ψj〉\n\n2\n\n〈sin 4ψj〉\n2\n\n〈sin 2ψj〉\n〈sin 4ψj〉\n\n2\n\n1− 〈cos 4ψj〉\n2\n\n\n\n\n−1\n\n×\n\n\n\n\n∑\nj Sj\n\n∑\nj Sj cos 2ψj\n\n∑\nj Sj sin 2ψj\n\n\n\n\n(4.11)\n\nwhere the hats indicate the maximum likelihood estimator maps, and 〈·〉 denotes\nthe average over all data samples j. We have also assumed that the noise variance\nis identical all detectors [33] and there is no correlation in time, nor variance of\nthe r.m.s noise with time between detectors.\n\nFollowing the equation 4.10, bandpass mismatch error maps are given by\n\n\n\n\nδÎBPM\n\nδQ̂BPM\n\nδÛBPM\n\n\n\n\n=\n\n\n\n\n1 〈cos 2ψj〉 〈sin 2ψj〉\n\n〈cos 2ψj〉\n1 + 〈cos 4ψj〉\n\n2\n\n〈sin 4ψj〉\n2\n\n〈sin 2ψj〉\n〈sin 4ψj〉\n\n2\n\n1− 〈cos 4ψj〉\n2\n\n\n\n\n−1\n\n×\n\n\n\n\nδ 〈Sj〉\n\nδ 〈Sj cos 2ψj〉\n\nδ 〈Sj sin 2ψj〉\n\n\n\n. (4.12)\n\nHere δ 〈Sj〉 , δ 〈Sj cos 2ψj〉 , and δ 〈Sj sin 2ψj〉 are sky component maps. We as-\nsumed perfect calibration of normalization of the CMB component for each detec-\ntor.\n\nFor future studies of the CMB polarization, the error of greatest concern arises\nfrom the leakage of the I component of the foregrounds into the Q and U compo-\nnents of the maximum likelihood band sky maps. From equation (4.12) we observe\nthat the three terms δ 〈Sj〉 , δ 〈Sj cos 2ψj〉 , and δ 〈Sj sin 2ψj〉 can potentially induce\na bias on the polarization Stokes parameters. The first term δ 〈Sj〉 has no impact\n\n\n\nResults 107\n\nif the maps of 〈cos 2ψ〉 and 〈sin 2ψ〉 vanish. This is the case in particular if the\ndetectors are arranged in sets of perfectly orthogonal pairs observing the sky. If in\naddition for each such pair there is a matching pair observing at an angle of 45◦\n\nrelative to the first one, we get an optimized configuration [33] for which the 3×3\nmatrix in equation (4.11) takes the form\n\n\n\n\n1 0 0\n\n0\n1\n\n2\n0\n\n0 0\n1\n\n2\n\n\n\n\n−1\n\n. (4.13)\n\nThis simple form is ‘optimized configuration’ of detectors orientation which was\nused for the Planck mission and this type of detector arrangement is now standard\nfor all proposed CMB polarization experiments. We then get\n\nδQ̂BPM(p) = 2δ 〈Sj cos 2ψj〉 ,\nδÛBPM(p) = 2δ 〈Sj sin 2ψj〉 , (4.14)\n\nAccording to the sky model presented in Sect. 4.1 we known that\n\nδ 〈Sj cos 2ψj〉 =\n∑\n\n(c)\n\nI(c)(p)\n∑\n\ni\n\nγ(c),ifi(p) 〈cos 2ψi,j〉 ,\n\nδ 〈Sj sin 2ψj〉 =\n∑\n\n(c)\n\nI(c)(p)\n∑\n\ni\n\nγ(c),ifi(p) 〈sin 2ψi,j〉 . (4.15)\n\nHere (c) labels the non-CMB components of the sky model and i labels the de-\ntectors. The coefficients γ(c),i parameters vary from detector to detector and it\ndepends on the gi(ν) function. fi(p) represents the fraction of the total hits in the\npixel p from the detector i, and 〈cos 2ψi,j〉 and 〈sin 2ψi,j〉 are the components of\nthe second-order crossing moments in pixel p for the detector i.\n\n4.2.1 Results\n\nIn this section, I present the numerical results for the bandpass mismatch error\nmaps and their angular power spectra using simulations. The exact correspon-\ndence of the pointing with the center of the pixel is an approximation which is\nperformed in order to isolate the effect of band-pass mismatch an to avoid intro-\nducing other effects. The timestreams for each detector is constructed by reading\n\n\n\nResults 108\n\na CMB map and a galactic thermal dust map Nside = 256. The value of Nside is\ncompatible with the proposed instrumental beam size. The input maps have been\npreviously convolved with a Gaussian kernel to account for the instrument beam\nθFWHM = 32′. We use an instrumental model with actual locations of detectors in\nthe focal plane as described in [79] or [35]. We have noticed that the details of the\narrangement of the detectors on the focal plane have little or no impact on the\nleakage angular power spectra. We simulate time streams by scanning input tem-\nplate maps without polarization, nor noise as well as same pixelization between\ninput and output maps using several detectors. We use detectors with nominal\nlocations in the focal plane and polarizer orientations for LiteBIRD. Because the\nmap making method is linear and the noise does not affect the bias induced by\nthe mismatch. For the same reason, we do not introduce sky emission polarization\nin simulations. The bandpass mismatch properties of each detector are generated\nrandomly and in a statistically independent manner. We use the pointing informa-\ntion in an ecliptic coordinate as well as a galactic coordinate. The hit map for 222\ndetectors and a year observation is shown in figure 4.6 in the galactic and ecliptic\ncoordinates. Scanning strategy creates symmetric pattern in ecliptic coordinates\nwith respect to angles θ, φ after one year observation. The simulation assumed\n222 detectors, which is the number of detectors composing the LiteBIRD arrays\ndescribed in [79], spread over a large focal plane approximatively 10 degrees wide\nobserving with no HWP. The observation time is a sidereal year length 365 days\nto ensure that the full and uniform sky is surveyed. We assume the fiducial scan-\nning parameters α = 65◦, β = 30◦, τspin = 10min, and τprec = 96.1803min for the\ncenter of the focal plane (see Section 4.2.3 for a discussion of the choice of τspin and\nτprec to minimize the inhomogeneity of the scanning pattern which is responsible\nfor Moiré effects in the crossing moment maps).\n\nFigure 4.6: Hitcount map of 222 detectors and 365 days for a fiducial scanning\nstrategy in the galactic and ecliptic coordinates.\n\n\n\nResults 109\n\nFigure 4.7 shows the Q and U leakage maps δQBPM and δUBPM for one particular\nrealization in galactic and ecliptic coordinate. The output polarization maps re-\nsult from optimal map making using our simulated noiseless and polarizationless\ntimestreams for the 140GHz channel. The bands at equal latitude visible in the\nleakage maps correspond to regions where the second order crossing moments de-\npart significantly from zero (Fig. 4.6), the strong correlation between the relative\nleakage amplitude and these moments will be demonstrated in the section 4.2.2.\n\nFigure 4.7: Q and U leakage maps, in galactic and ecliptic coordinates, with\nfiducial scanning parameters and Ndet = 222, a sidereal year survey.\n\nFigure 4.8 shows the mask of 20 % sky fraction which is the galactic plane. This\nsky fraction number is verified by the temperature angular power spectrum com-\nparison. Since we have intensity I component of the only CMB and the CMB plus\nthe thermal dust output map after applying the map making equation 4.9, we mask\nthe galactic plane and compute the angular power spectrum using \"healpy.anafast\"\nmethod. The result is plotted in the same figure with the angular power spectrum\nof the input CMB map. The sky fraction 20 % is a suitable masked sky fraction to\nhave the same amplitude of angular power spectra between the input and output.\nThe smaller sky fraction is tested by applying larger than 20 % masked galactic\nplane, the result is that the power spectrum is decreased as expected.\n\n\n\nResults 110\n\nFigure 4.8: Mask 20 % sky in galactic and ecliptic coordinate.\n\nFigures 4.9, 4.10 and 4.11 show the bandpass mismatch leakage contributions to\nthe EE and BB power spectra in different observing configurations. The power\nspectra are computed after masking the 20% of the sky where the thermal dust\nemission is the strongest. These plots also show the primordial power spectrum\nof BB mode model for two different values of tensor-to-scalar r = 10−2, 10−3. The\ndashed curves indicate how the signal is attenuated by convolution with a Gaussian\nbeam θFWHM = 32′. The power spectra are averaged of 10 realizations simulation.\n\nFigure 4.9 indicates that the bandpass mismatch error amplitude of the power\nspectrum scales as 1/Ndet the number of detectors. This scaling becomes more\naccurate when Ndet becomes large, as shown by comparing the EE and BB leak-\nage power spectra for τspin = 10min, τprec= 96.1803min and Ndet of either 74 or\n222. The pairs of spectra have the same shape but the ratio of power spectrum\namplitudes is consistent with the predicted ratio 222/74 = 3.\n\nFigure 4.10 shows the BB power spectra for α = 65◦, β = 30◦ for several spin\nand precession period ratio. We see that the characteristics of the leakage angular\npower spectrum particularly in the peaks locations at ` ≥ 100) depend on the\nexact ratio of τspin and τprec. A proper value of the ratio τprec/τspin moves the\npeaks in the bandpass leakage spectrum to higher `, away from the location of the\nmaximum of the primordial B-mode recombination bump. The more detail study\nwill be described in the section 4.2.3.\n\nFigure 4.11 compares the BB leakage power spectra for different configuration\nof scanning strategy of precession angles α and β, and also different spin period,\nprecession period. With the constraint α+β = 95◦, scan strategies with larger pre-\ncession angle produce less leakage because they allow for more homogeneous scan\n\n\n\nResults 111\n\nFigure 4.9: EE and BB leakage power spectra for α = 65◦, β = 30◦, τspin =\n10min, τprec= 96.1803min, and combining data for either 74 or 222 detectors.\nThe red curve corresponds to BB with 74 detectors, the cyan to EE with 74\ndetectors, the blue to BB with 222 detectors and the green to EE with 222\ndetectors. The purple curve represents a model of primordial B mode power\nspectrum with fiducial cosmological parameters after Planck for r = 0.01, the\nblack curves are including lensing for r = 0.01 and r = 0.001. The dashed curves\nshow the effect of convolving with a 32 arcmin beam. This plot demonstrates\n\nthe 1/Ndet dependance of the level of the power spectra.\n\nangle coverage per pixel, and hence lower |〈cos 2ψj〉| and |〈sin 2ψj〉| per individual\ndetector.\n\nWe observe that the power spectra above (without an HWP) are approximately\nproportional to `−η where η ≈ 2.5. We also observe some dependence of the\namplitude of the leakage spectra with respect to the scanning strategy parameters\nα and β. Scanning strategies with more uniform angular coverage (provided by\nlarger precession angles for the studied cases) have a lower leakage amplitude.\n\nOverall, the amplitude of the leakage due to bandpass mismatch error is nearly a\nGaussian with a zero mean and the variations of γd impact all multipoles of the\nleakage map power spectrum in a correlated way. For this reason, an accurate esti-\nmate of the average leakage power spectrum requires averaging many independent\n\n\n\nResults 112\n\nFigure 4.10: BB leakage power spectra for α = 65◦, β = 30◦,\nτspin=10min, τprec=93min (red); τspin=10min, τprec=96.1803min (green); and\nτspin=10/3min, τprec=96.1803min (blue). Simulations include 222 detectors and\n365 days observation. See the Fig. 4.9 caption for a description of the model\n\ncurves.\n\nrealizations even if many detectors are used for the simulations. At least on large\nangular scales, the fluctuations in the power spectrum due to different realiza-\ntions is roughly an overall amplitude varying as the square of a Gaussian. Figure\n4.12 shows 10 single realizations of the bandpass mismatch error, it indicates the\nfluctuation of the angular power spectrum for variations of γd.\n\nAs a conclusion, the study of bandpass mismatch error help to choose an opti-\nmal scanning strategy of future CMB polarization satellite. The amplitude of the\nbandpass leakage depends on the scanning strategy configuration of open preces-\nsion angle α and spin angle β, these peaks on the angular spectrum depends on\nthe ratio between precession period and spin period, the study is described in the\nsection 4.2.3. The sample rate does not impact to the leakage of the bandpass\nmismatch error, I already tested the case of the faster sampling rate, the result\nindicated that the amplitude of angular power spectrum is not affected. I have\nperformed many simulations with Nside = 256× 2 = 512 to verify the dependence\n\n\n\nResults 113\n\nFigure 4.11: BB leakage power spectra for different scanning parameters. In\ncyan: α = 65◦, β = 30◦, τspin=10min, τprec=96.1803min, red: α = 50◦, β =\n45◦, τspin=10min, τprec=96.1803min, green: α = 50◦, β = 45◦, τspin=2min,\nτprec=4 day, blue: α = 30◦, β = 65◦, τspin=2min, τprec=4 day. Spectra are\ncomputed for 222 detectors. Curves for the B mode model are described in\nFig. 4.9 caption. For the scanning strategies with a long precession period, we\ncomputed spectra for 100 detectors rescaling to 222 equivalent detectors using\n\nthe 1/Ndet dependance.\n\nwith resolution of those results. The results are similar. The best sampling rate\nchoice could be determined by studying other systematic effects such as 1/f noise\nperformance, cosmic rays interaction with detectors.\n\nTable 4.1 shows the contribution of bandpass miss match error leakage to tensor-\nto-scalar r based on its angular power spectrum averaged over many realizations.\nThe calculation is using chi-square estimation:\n\nδ̂r =\n\n∑`max\n\n`=`min\n(2`+ 1)C`Ĉ`∑`max\n\n`=`min\n(2`+ 1)C2\n\n`\n\n. (4.16)\n\nHere C` is the power spectrum for the primordial B mode signal normalized to\nr = 1. Ĉ` is the angular power spectra signal due to the bandpass mismatch\nleakage. The Table shows δr calculated for two ranges of `: one with ` ∈ [2, 10] to\n\n\n\nResults 114\n\nFigure 4.12: The angular power spectrum of 10 realizations with different set\nof γd parameter.\n\n2 ≤ ` ≤ 10 10 ≤ ` ≤ 200\nα = 30◦; β = 65◦; τprec = 4days; ωspin = 0.5 rpm 1.83 ×10−3 9.32 ×10−5\n\nα = 50◦; β = 45◦; τprec = 4days; ωspin = 0.5 rpm 6.49 ×10−4 4.66 ×10−5\n\nα = 50◦; β = 45◦; τprec = 96min; ωspin = 0.1 rpm 6.32 ×10−4 3.08 ×10−5\n\nα = 65◦; β = 30◦; τprec = 93min; ωspin = 0.1 rpm 3.29 ×10−4 7.61 ×10−5\n\nα = 65◦; β = 30◦; τprec = 96min; ωspin = 0.1 rpm 3.27 ×10−4 2.11 ×10−5\n\nα = 65◦; β = 30◦; τprec = 96min; ωspin = 0.3 rpm 3.03 ×10−4 1.77 ×10−5\n\nTable 4.1: Contribution of bandpass mismatch error to the tensor-to-scalar\nratio r computed according to the equation (4.16) for 222 detectors and 365\ndays observation. The level of the bandpass leakage relative to primordial B-\nmode signals is acceptable at the angular scale of the recombination bump, but\nproblematic in the reionization bump at ` ≤ 10. Scanning strategies with larger\nα and smaller β perform better, as they provide more uniform angular coverage\n\nin each pixel.\n\nisolate the signal from the re-ionization bump, and another with ` ∈ [10, 100] to\nisolate the signal arising from the recombination bump. The results in the table\nassumeNdet = 222 detectors but can be rescaled based on the 1/Ndet dependence to\nother numbers of detectors. These results are only an order of magnitude estimate\nbecause they are based on a single 140GHz channel, and it has been assumed\n\n\n\nResults 115\n\nthat very low and very high-frequency channels have been used to remove the\nnon-primordial components completely. We stress that the bandpass mismatch\npower spectrum is not a simple bias that can be predicted and subtracted away\nbecause its overall amplitude suffers large fluctuations, which is of the same order\nof magnitude as the average bias itself.\n\nThe adopted model for the band-passes uncertainty assumes a 1 % deviation from\nnominal value of νmin and νmax. I checked how the result scales with the value of\nthis uncertainty, moving this for example to 2%. I have verified that the resulting\nγd are increased by a factor of 2, as was predicted. The result in figure 4.13 shows\nthat the angular power spectrum is scaled by a factor of 4. The simulation used\nthe fiducial scanning strategy and 10 realizations.\n\nFigure 4.13: The angular power spectrum of 2 % variation of the filters, the\npower spectrum is scaled by 4.\n\nOne more point, the errors on the band definition are a uniformly distributed.\nWhat if there is a global offset or some systematic variation across the focal\nplane? This is important but difficult to address in absence of instrumental mod-\nels of the origin of bandpass mismatches. In the most pessimistic case, there\nis a constant difference between detector γa and detector γb coherent across the\n\n\n\nResults 116\n\nfocal plane as shown in figure 4.14, where the \"a\" bolometers are the ones for\nwhich the orientation of the polarizer is closer to the horizontal axis in the de-\ntector frame. In that case, we observe that the leakage is boosted by a large\nfactor compared to the uncorrelated case resulted in figure 4.15. This is be-\ncause the leakages for each pair do not tend to cancel each other in the global\nleakage map including all detectors. However as we mentioned before we use\ndetectors with nominal locations in the focal plane and polarizer orientations,\neach detector has a different value of γd, these pairs are orientated following\n(0◦, 90◦), (−45◦, 45◦), (−120◦,−30◦), (−165◦,−75◦), (−180◦,−90◦), (−225◦,−135◦)\n\non the focal plane. Figure 4.16 presents the leakage bandpass mismatch error maps\nfor each pair orientation for a year observation and fiducial configuration scan. We\ncan observe obviously that there have negative and positive patterns on the leak-\nage maps of several pair orientations. Therefore there have cancellation across the\nfocal plane when we use multi-detectors, this also implies the scale of 1/Ndet in\nthe amplitude of power spectrum.\n\nFigure 4.14: The global offset across the focal plane\n\nFigure 4.17 shows that the amplitude of the angular power spectrum of each pair\nwith nominal locations on the focal plane is the same as expected.\n\nI have also computed the bandpass errors in case of Planck scanning strategy\nα = 7.5◦, β = 85◦, τspin = 1min, τprec= 6month. Figure 4.18 shows the polarized\nleakage maps for 222 detectors and a sidereal year observation. In order to compare\nthe level of the bandpass mismatch systematic error of the Planck satellite scanning\nstrategy and the fiducial scanning strategy, the angular power spectra are plotted\nin figure 4.19. We observe that the leakage is higher by more than an order of\nmagnitude in the power spectrum. It is obviously understandable because of the\nsmall opening angle α used to measure temperature anisotropies of Planck mission.\n\n\n\nResults 117\n\nFigure 4.15: Angular power spectrum of the global offset across the focal plane\ncompares with the nominal focal plane.\n\nFigure 4.16: The leakage map of pair detectors with nominal locations. The\ntitle indicates the oriental angle of a pair detector.\n\n\n\nResults 118\n\nFigure 4.17: The BB angular power spectrum for each pair detector with\nnominal locations, the orientation of pair detector is labeled.\n\nThanks to the fast scanning with the Planck mission, we do not observe peaks on\nthe high multipoles on the angular power spectrum.\n\nSo far, I presented the results of the case of no half-wave plate. Future studies will\nconsider an imperfection of a rotating half-wave plate case. In case of a perfect\nHWP, we perform a simple set of simulations in which the input sky (smoothed\nby a 32′ beam) is a HEALPix map pixelized at nside = 256. The pixel size is well\nmatched to the rotation speed of the HWP, which makes about one turn while it\ncrosses a pixel. However, numerical effects will generate unevenness in the angular\ncoverage of each pixel, and thus, when multi-detector maps are made using the\nequation 4.9, small bandpass leakage mismatch effects will subsist. Simulating the\nobservation of this model sky with the use of a HWP spinning at 88 rpm and other\nparameters set to α = 65◦, β = 30◦, τspin=10min, τprec = 96.1803min, we obtain\nthe small residual leakage shown in figure 4.20, which confirms the effectiveness of\nthe HWP in reducing bandpass leakage by homogenizing the angular coverage in\neach pixel. The shape of the spectrum of the residual is similar to that of white\nnoise. Its origin is in the small unevenness of the angle distributions across the\n\n\n\nAnalytic estimates 119\n\nFigure 4.18: The hitcount map and leakage maps of the Planck scanning\nstrategy case, α = 7.5◦, β = 85◦, τspin = 1min, τprec= 6month, and combin-\ning data for 222 detectors and 365 days observation a single realization. The\npower spectrum of the fiducial scanning strategy is also plotted with purpose of\n\ncomparison.\n\npixels and is an artefact of sky pixelization.\n\nI verified that in case of a perfect HWP, the multi-detector solution for the polar-\nization is close to the solution consisting in combining single detector polarization\nmaps, as the residual leakage and its impact of r that can be read off the plot, is\nnegligible.\n\n4.2.2 Analytic estimates\n\nIn order to understand the features in the leakage maps related to the scanning\nstrategy configuration. The correlation of leakage amplitude and crossing moment\n〈cos 2ψ〉 and 〈sin 2ψ〉 provides an easy and fast way to predict the magnitude of\nleakage. We now consider the signal of the leakage from a pair i detector Si;a(t)\n\n\n\nAnalytic estimates 120\n\nFigure 4.19: Power spectrum of the Planck scanning strategy case which is\ndescribed in figure 4.18, I also plotted the plot of the fiducial scanning strategy\nis α = 65◦, β = 30◦, τspin = 10min, τprec= 96min, and combining data for 222\n\ndetectors and 365 days observation\n\nand Si;b(t) at time t in pixel p with no noise assumption.\n\nSi;a(t) = Ii;p +Qp cos 2ψ(t) + Up sin 2ψ(t) +Mi;p,\n\nSi;b(t) = Ii;p −Qp cos 2ψ(t)− Up sin 2ψ(t)−Mi;p. (4.17)\n\nHere ψ is the polarizer angle of detector a and Ii;p, Qp, Up are the Stokes param-\neters. The bandpass mismatch component Mi;p is given by\n\nMi;p =\n1\n\n2\n\n∑\n\n(c)\n\n(\nγa\n\n(c) − γb\n(c)\n\n)\nIp,(c). (4.18)\n\nHere (c) denotes for foreground components. As mentioned before, the\n(\nγa\n\n(c) − γb\n(c)\n\n)\n\nis vary from detector pair to detector pair. Each pair detector is orthogonality then\n〈cos 2ψ〉 and 〈sin 2ψ〉 vanish.\n\n\n\nAnalytic estimates 121\n\nFigure 4.20: EE and BB leakage power spectra with rotating HWP\nfor α = 65◦, β = 30◦ and spin period of 10min with a HWP rotating at 88 rpm\n\nfor 50 detectors.\n\nFrom the equation 4.12, the map making equation of bandpass mismatch error for\na pair is given by\n\n\n\n\nÎp\n\nQ̂p\n\nÛp\n\n\n =\n\n\n\n\n1 0 0\n\n0 1\n2\n\n(1 + 〈cos 4ψ〉) 1\n2\n〈sin 4ψ〉\n\n0 1\n2\n〈sin 4ψ〉 1\n\n2\n(1− 〈cos 4ψ〉)\n\n\n\n\n−1\n\n\n〈S〉\n〈1\n\n2\n(Sa − Sb) cos 2ψ〉\n〈1\n\n2\n(Sa − Sb) sin 2ψ〉\n\n\n .\n\n(4.19)\n\nWe neglected the index of detector label i with purpose of simplification. The\nestimated Stokes parameter maps Q̂p and Ûp can be decomposed as Q̂p = Qp+δQp\n\nand Ûp = Up + δUp, where δQ and δU represent the leakages to polarization\nresulting from bandpass mismatch. The leakage maps of intensity to polarization\n\n\n\nAnalytic estimates 122\n\nis described by\n\n(\nδQp\n\nδUp\n\n)\n=\n\n\n\n\n1\n\n2\n(1 + 〈cos 4ψ〉) 1\n\n2\n〈sin 4ψ〉\n\n1\n\n2\n〈sin 4ψ〉 1\n\n2\n(1− 〈cos 4ψ〉)\n\n\n\n\n−1(\n〈Mp cos 2ψ〉\n〈Mp sin 2ψ〉\n\n)\n\n=\n2\n\n(1− 〈cos 4ψ〉2 − 〈sin 4ψ〉2)\n\n(\n1 + 〈cos 4ψ〉 −〈sin 4ψ〉\n−〈sin 4ψ〉 1− 〈cos 4ψ〉\n\n)\n\n×\n(\n〈Mp cos 2ψ〉\n〈Mp sin 2ψ〉\n\n)\n. (4.20)\n\nWe can assuming the average distribution angles of 〈cos 4ψ〉2 + 〈sin 4ψ〉2 � 1.\nThis assumption is not so bad an approximation except very near the poles, then\nwe obtain a relationship between leakage maps and distribution angles\n\n(\nδQp\n\nδUp\n\n)\n≈ 2\n\n(\n〈Mp cos 2ψ〉\n〈Mp sin 2ψ〉\n\n)\n. (4.21)\n\nSubstituting Mp expression in the equation 4.18 and dividing the leakage map to\nthe galactic components, we obtain the correlation function of the amplitude of\nthe leakage and the distribution angle\n\n(\nδQp\n\nIGal;p\n\nδUp\n\nIGal;p\n\n)\n=\n(\nγa\n\nGal − γb\nGal\n\n)\n(\n〈cos 2ψ〉\n〈sin 2ψ〉\n\n)\n. (4.22)\n\nWe can define that these distribution angles 〈cos 2ψ〉 , 〈sin 2ψ〉 are crossing mo-\nments of a single detector. With the help of simulation for a pair detector I have\nverified the relationship of leakage maps and crossing moments. Figure 4.21 shows\nthese maps in the equation 4.22 the relative leakage map δQp/IGal;p and the cross-\ning moment map\n\n∑\ncos 2ψ/np. The δUp/IGal;p and\n\n∑\nsin 2ψ/np components have\n\nsimilar properties.\n\nThe tight link of the leakage map and the crossing moment map due to bandpass\nmismatch error is shown in figure 4.22 using scattering histogram two dimensions\nplot for a subset of pixels of these maps. We observed that the high correlation of\ntwo maps has a linear slope following ∆γ = γa−γb as the equation 4.22. It means\nthat the approximations made to derive equation 4.22 are verified. The exception\nof the linear slope dues to pixels near to poles where the angle coverage of crossing\nmoment is less uniform for the fiducial scanning strategy.\n\n\n\nAnalytic estimates 123\n\nFigure 4.21: Left: Leakage for the Q component relative to the dust temper-\nature (δQ/IGal) after polarization reconstruction using one bolometer pair only\nand a one year observation time. Right: Averaged cos 2ψ in each pixel for one\nbolometer after one year of observation time. This quantity is strongly correlated\n\nto the relative leakage Q component with respect to the dust intensity.\n\nFigure 4.22: Values of the relative leakage δQp/IGal;p for a pair of detectors\nwith orthogonal polarizations of a function of the scanning strategy parameter\n(1/np)\n\n∑\ncos 2ψ (see text) after map making with two detectors only. We observe\n\na tight correlation between the relative leakage and the second order crossing\nmoments.\n\nWe can develop the equation 4.22 to find the solution combining many detectors.\nWe also start from the equation 4.20, the leakage component of multi-detector is\n\n\n\nAnalytic estimates 124\n\ngiven by\n\n(\nδQp\n\nδUp\n\n)\n=\n\n\n\n\n1\n2\n\n∑\ni\n\n∑\nj\n\n(1 + cos 4ψj\ni)\n\n1\n2\n\n∑\ni\n\n∑\nj\n\nsin 4ψji\n\n1\n2\n\n∑\ni\n\n∑\nj\n\nsin 4ψj\ni\n\n1\n2\n\n∑\ni\n\n∑\nj\n\n(1− cos 4ψji )\n\n\n\n\n−1\n\n\n∑\ni\n\n∑\nj\n\ncos 2ψj\ni Mi,p\n\n∑\ni\n\n∑\nj\n\nsin 2ψj\ni Mi,p\n\n\n\n\n(4.23)\n\nHere we labeled the detector pairs i and all falling samples j in pixel p. Since the\nnumber of detector pair is increased, the covariance matrix in the equation 4.23\nbecomes closely diagonal. So that we obtain a simple approximation:\n\nδQp\n\nIGal;p\n\n≈ 2\n\nNhit\n\n∑\n\ni\n\n∆γi\n\n∑\n\nj\n\ncos 2ψj\ni , (4.24)\n\nHere Nhit is the total number of hit counts of all detectors and ∆γi = γai − γbi .\nBecause γ parameter is random and independently, we can express variance of the\nequation 4.24 into separated terms\n\nVar\n\n(\nδQp\n\nIGal;p\n\n)\n≈\n∑\n\ni\n\nVar(∆γi)\n\n(∑\n\nj\n\ncos 2ψj\ni\n\n)2(\n2\n\nNhit\n\n)2\n\n, (4.25)\n\nwe have Var(∆γ) = 2Var(γ) then we obtain the variance of the leakage maps and\ncrossing moment cosine in case of multi-detectors is\n\nVar\n\n(\nδQp\n\nIGal;p\n\n)\n≈ 4\n\nVar(γ)\n\nNdet\n\n〈(∑\ncos 2ψji\nn̄p\n\n)2〉\n\ndet\n\n, (4.26)\n\nHere 〈 · 〉det means average over all detectors, and n̄p = Nhit\n\nNdet\nis definition of the\n\naverage number of hits per detector. The equation 4.26 is the expression for\nQ component, the U component has similar variance of leakage map with the\ncrossing moment sine map. The comparison of variance leakage map the quan-\ntity Var (δQp/IGal;p) and crossing moment the quantity\n\n〈(\n(1/n̄p)\n\n∑\ncos 2ψi\n\n)2〉\ndet\n\nshows in the figure 4.23. The result is simulated by 10 times Monter Carlo real-\nizations.\n\nSimilarly in case of a pair detector, figure 4.24 shows the correlation of the equation\n4.26 in the scatter plot. We observed that the significant correlation between two\nquantities is especially on large scales. The dispersion is partly due to the limited\nnumber of realizations to estimate the variance. Nevertheless, the result shows that\n\n\n\nPrecession period and spin period ratio τprec/τspin 125\n\nFigure 4.23: Left: Estimated leakage variance of the Q component relative\nto the dust temperature (Var (δQp/IGal;p)) after polarization reconstruction us-\ning all bolometer pairs and one year of observations. We used 10 indepen-\ndent realizations of the bandpass to estimate the variance. Right: Averaged〈(\n\n(1/n̄p)\n∑\n\ncos 2ψji\n\n)2〉\ndet\n\nin each pixel for all bolometers after one year of ob-\nservation time. As for the detector pair case, we observe a tight correlation\n\nbetween the two maps on large angular scales\n\nwe can evaluate the level of bandpass mismatch error by study only the second\norder crossing moments maps cosine and sine without the need of running large\nsimulations of map making. This method also explains the result in figure 4.11,\nit means that the scanning strategies ( α and β parameters) with more uniform\nangle distribution (larger precession angle α) have lower the impact of bandpass\nmismatch error (see also [150] for the link with other systematic effects).\n\n4.2.3 Precession period and spin period ratio τprec/τspin\n\nThe observing strategy of a CMB polarization mission must be designed to cover\nthe complete sky with adequate polarization angle coverage at each point. Ho-\nmogeneous observing time among all pixels and all polarization angles provides\nthe lowest noise power spectrum, uncorrelated between maps of I, Q, and U. The\nhomogeneity of the sky and angle coverage can be optimized with a proper choice\nof the values of the various angles, and with an optimization of the spinning and\nprecession period. In this section, we discuss how to optimize precession and spin\nperiod scanning parameters.\n\nMoiré effects in the leakage maps, generating peaks in the bandpass leakage angular\npower spectrum, arise from concentrations of scans in particular regions of sky and\nwith sharp discontinuities in the angle coverage maps. The most notable effect\narises for τspin = 10 min and τprec = 93min, α = 65◦ and β = 30◦ as shown in\n\n\n\nPrecession period and spin period ratio τprec/τspin 126\n\nFigure 4.24: Estimated variance distribution of the relative leakage parameter:\n\nVar (δQp/IGal;p) as a function of c =\n〈(\n\n(1/n̄p)\n∑\n\ncos 2ψt;i\n\n)2〉\ndet\n\n(see text) after\nmap making including all detectors. We have averaged over ten realizations to\n\nestimate the variance.\n\nFigure 4.25: Effect of a poorly chosen scanning frequency ratios. left The map\nhas τprec/τpin = 9.3, right the map has the more irrational ratio τprec/τpin =\n9.61803,. A series of Moiré patterns on intermediate angular scales is clearly\nvisible in the map on the left, which lead to spikes in the crossing moment map\npower spectra, and also in the final bandpass mismatch power spectra. The\nartefacts can be avoided by choosing ratios of frequencies judiciously in order to\n\navoid good rational approximations.\n\nfigure 4.10. We can adjust τprec/τspin concerning to crossing moment while the sky\n\n\n\nPrecession period and spin period ratio τprec/τspin 127\n\nis scanned. The figure 4.25 shows crossing moment maps for two different ratios\nof precession period and spin period ratio.\n\nFigure 4.26: The hit count map of a pair detector in a assumed pessimistic\nconfiguration: τspin = 10min, τprec= 89.95min. The time integration of each\nscan running for 1 day, 6 days, 1 month, 2 months, 4 months, 6 months, 8\n\nmonths and 1 year, from left to right and top to bottom, respectively.\n\n\n\nPrecession period and spin period ratio τprec/τspin 128\n\nFigure 4.27: The hit count map of a pair detector for the fiducial scanning\nstrategy α = 65◦, β = 30◦, τspin=10min, τprec = 96.1803min. The time inte-\ngration of each scan running for 1 day, 6 days, 1 month, 2 months, 4 months, 6\nmonths, 8 months and 1 year, from left to right and top to bottom, respectively.\n\n\n\nPrecession period and spin period ratio τprec/τspin 129\n\nAs an example, in a assumed pessimistic case, we choose a bad ratio of param-\neter of precession and spin period, as τspin = 10min, τprec= 89.95min. Figure\n4.26 and figure 4.27 illustrate the evolution of scanning hit count maps of a pair\ndetector for the pessimistic case τspin = 10min, τprec= 89.95min and the figucial\nconfiguration α = 65◦, β = 30◦, τspin=10min, τprec = 96.1803min. The fiducial\nconfiguration has been chosen a good ratio of the precession period and the spin\nperiod, τprec/τspin which follows an irrational number as Golden ratio, so that the\nhit count maps present homogeneous angle coverage. In contrast, the hit count\nmaps of the assumed pessimistic configuration have many inhomogeneous features.\n\nIn order to observe these feature in the pessimistic case in the leakage maps as\nwell as in the final angular power spectrum compare with the fiducial scanning\nstrategy, the simulation implements for the configuration α = 65◦, β = 30◦, τspin =\n10min, τprec= 89.95min, 222 detectors and a year observation, these leakage maps\nare also obvious features in the hit count maps. Figure 4.28 shows the hitcount\nmap and leakage maps of the assumed pessimistic case in the galactic coordinate.\nFigure 4.29 indicates that the amplitude of the power spectrum is as same as the\n\nFigure 4.28: The hit count and leakage maps of a bad chosen parameter of\nthe precession time, the scanning parameters are α = 65◦, β = 30◦, τspin =\n10min, τprec= 89.95min in the galactic coordinate, and combining data for 222\n\ndetectors and 365 days observation for one realization\n\nfiducial scanning strategy at the reionization bump. However the feature in the\n\n\n\nPrecession period and spin period ratio τprec/τspin 130\n\nFigure 4.29: Power spectrum of the badly chosen parameter of precession pe-\nriod, the scanning strategy parameters are described in figure 4.28. The power\nspectrum of the fiducial scanning strategy is also plotted together with the pur-\n\npose of comparison.\n\nhitcount maps (it implies crossing moment maps) produces booting in the power\nspectrum at the recombination bump.\n\nIn order to investigate the position of peaks on the power spectra on the multipoles\n` axis. The scanning angles are fixed α = 65◦ and β = 30◦, with the same 222\ndetectors and 365 days observation. Firstly the spin period is fixed, the precession\nperiod is variation from ∼ 96.1803min to ∼ 101.1803min. The simulations have\ntake into account the float number of the scanning parameter. Figure 4.30 shows\nthe result of power spectra for different precession periods, the amplitude of each\nconfiguration is rescaled by order of magnitudes with the purpose of comparison.\nWe observed that these positions of peaks tend to shift to smaller multipoles `,\nthe number of peaks also have variation in the power spectrum depending on the\nratio of precession and spin period. These hitcount maps leakage maps are not\nshown here but these power spectra mean these maps have different features.\n\nSecondly, the spin period is varied including a tiny change with many numbers after\n\n\n\nPrecession period and spin period ratio τprec/τspin 131\n\nFigure 4.30: Variation of precession time parameter and the amplitudes are\nrescaled to easily observe. The other canning strategy parameters are fixed as\nα = 65◦, β = 30◦, τspin = 10min, 222 detectors and a year of observation. The\n\nvalue of precession period is labeled on the plot.\n\nthe dot of a float number fixing other parameters. With the same configurations,\nwe observed that the position and amplitude of these peaks are not affected since\nthe spin period is smaller than 0.3 rpm. In the case of spin period is 0.5 rpm these\npeaks are moved away as shown in figure 4.31. Thirdly the ratio of precession\nperiod and spin period τprec/τspin is varied for example precession is 32.06 minutes\nand the spin period is 0.3 rpm. Figure 4.31 shows that these peaks are located at\nthe same position on the multipoles axis. Since the ratio is varied if we change the\nspin or precession period, the location of these peaks is changed. We also applied\na tiny change in the float number of the spin period, the result shows that there\nis a change in the amplitude of peaks.\n\nAs a conclusion, to obtain good crossing moment maps (it means lower the power\n\n\n\nPrecession period and spin period ratio τprec/τspin 132\n\nFigure 4.31: Variation ratio of precession time and spin parameters. The\namplitudes are rescaled to easily observe. The canning strategy parameters are\nfixed α = 65◦, β = 30◦. (from bottom to the top) The blue green red and\ncyan curves have different spin period from τ = 0.1 rpm up to τ = 0.5 rpm.\nThe violet and yellow curves are the power spectra in case of changing ratio of\n\nprecession and spin period together.\n\nspectrum also), we have to carefully consider the ratio of scan frequencies of pre-\ncession and spin period. Those peaks at multipole numbers on the bandpass\nmismatch power spectrum are caused by Moiré effects. The detailed study of this\nratio jumps to the number theory of regular and irregular motion3. The first study\nwas carryed out for τspin = 10min and τprec = 93min and we obtain the hitcount\nand crossing moment maps with clearly visible Moiré effects as shown in the figure\n4.25 on the left. After that the map with ratio τprec/τspin = 9.61803, has the more\n3See detail discussion in Michael Berry, (1978, September), Regular and irregular motion, in S.\nJorna (Ed.), AIP Conference proceedings (Vol. 46, No. 1, 16-120), the discussion of perturbed\nintegrable systems in classical mechanics and the stability of the solar system.\n\n\n\nA correction method 133\n\nirrational number. The detailed description of the importance of avoiding reso-\nnances and theory of choosing ratios number theory is described in the published\npaper of this bandpass mismatch error for future CMB satellite experiment [50].\n\n4.3 A correction method\n\nThere have several correction methods which are studied in the companion paper\n[14]. In this section, I present the study of correction method for a detector pair\nwhich is difined in equation 4.17. The procedure has been used for the WMAP data\nanalysis [63] which cancels leakage form intensity to polarization due to bandpass\nmismatch. We assume perfectly uniform angle coverage in each pixels for more\nthan three revisit. We also assume white noise with identical variance for a pair.\nThe method consists in estimating an extra component as the leakage on a pair by\npair basic. This is equivalent to estimating I, Q, U components for each detector\nand then combine the Q and U parameters obtained for each detector.\n\nCov3;p = (ATN−1A)−1\n\n=\nσn\nNp\n\n×\n\n\n\n\n1 〈cos 2ψ〉 〈sin 2ψ〉\n\n〈cos 2ψ〉 1 + 〈cos 4ψ〉\n2\n\n〈sin 4ψ〉\n2\n\n〈sin 2ψ〉 〈sin 4ψ〉\n2\n\n1− 〈cos 4ψ〉\n2\n\n\n\n\n−1\n\n. (4.27)\n\nIn the second case the polarization covariance matrix is the submatrix covariance\nwhich is formed by the lower tight corner of 4.27. Here σn is the root mean square\nof noise in each pixel.\n\nCov2;p = σn ×\n\n\n\n\n1 + 〈cos 4ψ〉\n2\n\n〈sin 4ψ〉\n2\n\n〈sin 4ψ〉\n2\n\n1− 〈cos 4ψ〉\n2\n\n\n\n\n−1\n\n. (4.28)\n\nWe study the loss of accuracy in two cases by computing numerically the two\ncovariance matrices with and without leakage components. In case of scanning\nstrategy α = 65◦, β = 30◦, τspin = 0.1 rpm the noise of the covariance matrices\nassociated with the level of leakage is about 4 %. In case of α = 50◦, β = 45◦, the\nloss accuracy of the Q component is of the order of 10 % for a given arbitrary\n\n\n\nA correction method 134\n\nFigure 4.32: Histogram of the covariance matrix and sub-covariance matrix\nfor the configuration α = 30◦, β = 65◦, τspin = 2 min, τprec = 4 day.\n\nFigure 4.33: Histogram of the covariance matrix and the sub-covariance matrix\nfor the configuration α = 50◦, β = 45◦, τspin = 2 min, τprec = 4 day.\n\n\n\nA correction method 135\n\nFigure 4.34: Histogram of the covariance matrix and the sub-covariance matrix\nfor the configuration α = 50◦, β = 45◦, τspin = 10 mins, τprec = 96.1803 mins.\n\nFigure 4.35: Histogram of the covariance matrix and the sub-covariance matrix\nfor the configuration α = 65◦, β = 30◦, τspin = 10 min, τprec = 96.1803 minutes.\n\n\n\nQUBIC TES 136\n\ndetector pair. These histograms in figure 4.32, 4.33, 4.34, 4.35 carry out for\ndifferent configurations of scanning strategy parameters. QQ and UU polarization\nmaps in case of temperature leakage components are estimated as equation 4.27.\nQQsub and UUsub are in case of null leakage as equation 4.28. The histogram of\ninverse of the covariance matrices respectively\n\n1\n\nQQ\n,\n\n1\n\nQQsub\n\n,\n1\n\nUU\n,\n\n1\n\nUUsub\n\n. The root\nmean square of noise σn = 1.\n\nAs a conclusion, if we assume, there is no leakage, we will estimate the sub-\ncovariance matrix Q, U. In contrast in case of leakage we will estimate the level\nleakage of a pair similar to a single detector with covariance matrix I, Q, U.\nThis is estimation of the total budget of noise (3 components I, Q, U) of leakage\ncomponents of a single detector maps as equation 4.27 comparing with null the\nleakage assumption of the detector pair which has the orthogonal polarization as\nequation 4.28. The comparison of the full focal plane is left for a future study.\nThe second method which is base on a template fitting method is studied in the\ncompanion paper [14].\n\n4.4 Conclusions\n\nThis chapter described estimation of the level of bandpass mismatch error to the\nfinal determination of power spectrum as well as tensor-to-scalar value r for several\nscanning strategies. This study helps to choose an optimal scanning strategy for\nfuture CMB polarization satellite. In case of without an HWP, bandpass mismatch\nerror is a non-negligible systematic effect. The amplitude of bandpass mismatch\nerror depends on the scanning strategy of the satellite: precession angle α, spin\nangle β, precession spin τprec and rotating spin τspin. The amplitude scales as the\nnumber of detectors. I found that the spurious angular power spectrum could\npotentially bias r for measurements of the reionization bump (2 ≤ ` ≤ 10) at the\norder of about 5 × 10−4, and of the recombination bump (10 ≤ ` ≤ 200) at the\norder of about 5×10−5 depending on scanning angle parameters and for variations\nof band-pass observed by Planck. The amplitude of the power spectrum is scaled\n\n1\n\nNdet\n\nthe number of detectors. I observed a tight correlation between leakage maps\n\nand the crossing moment, 〈cos 2ψ〉, 〈sin 2ψ〉, this is a fast and easy way to predict\nthe magnitude of potential leakage. The companion paper [14] presents correction\nmethods for bandpass miss match error [in preparation].\n\n\n\nChapter 5\n\nInteraction of particles with a TES\n\narray\n\nContents\n5.1 Theory of a superconducting Transition-Edge Sensor . 139\n\n5.1.1 Theory of superconductivity . . . . . . . . . . . . . . . . 139\n\n5.1.2 The superconducting transition region . . . . . . . . . . 141\n\n5.1.3 Principle of a Transition-Edge Sensor (TES) . . . . . . . 142\n\n5.1.3.1 Electrical and thermal response . . . . . . . . . 142\n\n5.1.3.2 Noise performance . . . . . . . . . . . . . . . . 151\n\n5.2 TES arrays of the QUBIC experiment . . . . . . . . . . 155\n\n5.3 The cryostat and the electronic readout system . . . . 160\n\n5.3.1 IV, PV, RV curves . . . . . . . . . . . . . . . . . . . . . 173\n\n5.4 Radioactive source Americium 241 . . . . . . . . . . . . 176\n\n5.5 TES model approach . . . . . . . . . . . . . . . . . . . . 177\n\n5.6 Glitches data analysis . . . . . . . . . . . . . . . . . . . . 182\n\n5.6.1 Glitches detection . . . . . . . . . . . . . . . . . . . . . 184\n\n5.6.2 Fit glitches . . . . . . . . . . . . . . . . . . . . . . . . . 190\n\n5.6.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . 195\n\n5.6.4 Time constants and theKI parameter of the PID controller198\n\n5.6.5 Time constants, amplitude and the voltage bias VTES . 201\n\n5.7 Cross-talk . . . . . . . . . . . . . . . . . . . . . . . . . . . 203\n\n137\n\n\n\nTheory of a superconducting Transition-Edge Sensor 138\n\n5.7.1 Thermal cross-talk . . . . . . . . . . . . . . . . . . . . . 203\n\n5.7.2 Cross-talk of the electronic readout chain . . . . . . . . 209\n\n5.8 Conclusion and discussion . . . . . . . . . . . . . . . . . 211\n\nWe know from the Planck mission that the cosmic ray particles interaction with\nthe focal plane produces thermal glitches on the data due to deposit energies.\nExamples of Planck data are shown in figure 3.17, the cosmic rays systematic is\ndescribed in the chapter 3 the section 3.9. For future CMB missions, we need to\nstudy this systematic effect and find a way to reduce its impact.\n\nQUBIC is a ground-based experiment, aiming at measuring B-mode polarization of\nthe CMB. The QUBIC instrument design is based on a novel concept of bolometric\ninterferometry with high sensitivity and extremely good control of the systematic\neffects by the self-calibration technique. The QUBIC focal plane contains a total\nof 2048 pixels Transition-Edge-Sensors (TES) in the focal plane. A TES, also\ncalled a superconducting phase-transition thermometer, consists of a supercon-\nducting film operated in the narrow temperature region between the normal and\nsuperconducting state. The concept of QUBIC is described in the section 3.10.1.\nIn the \"mm lab\" of APC laboratory we set up an Americium 241 source in front\nof a QUBIC’s 256-TES detector array inside the cryostat, which can cool down to\nthe level of hundreds of mK. When a particle hits components of a TES pixel(e.g.:\nThermometer, absorbing grid membrane, substrate), the deposited energy could\nbe transformed to temperature elevation between components and to the neighbor\npixels (called cross-talk). This study does not only help to understand the TES\nthermal behaviors but also prepares for future CMB missions. In addition, the\nstudy of cross-talk for TESs pixel array also carry out. The idea is that when we\nhave a glitch event in the signal, we also detect the behavior on pixels around.\n\nIn this chapter, I am going to describe (1) the theory of superconductivity associ-\nated with the transition regime, (2) principles of a superconducting transition edge\nsensor, I figure out the electrical time constant, the thermal time constant basing\non a TES responding equations, noise performance in a TES. (3) I also describe\nthe TES array and the electronic readout system of the QUBIC experiment, the\ntime constant of the electronic readout chain. (4) In the end, I present my work\non data analysis of glitches and the cross-talk.\n\n\n\nTheory of a superconducting Transition-Edge Sensor 139\n\n5.1 Theory of a superconducting Transition-Edge\n\nSensor\n\nIn a normal conductor/semiconductor the current is carried by electrons (i.e.\nFermions) which obey Fermi-Dirac statistics while in a superconductor the current\nis carried by Cooper-pairs (i.e. Bosons) which obey Bose-Einstein statistics. The\nprinciple is that the phonon lattice net slows down the velocity of electrons so that\nthe electrons joined into Cooper-pairs. The development of a sensitive supercon-\nducting detector allows us to measure a power source with faster responses and a\nlarger heat capacity [61]. A superconducting detector works at low temperature\nin that case the noise level is reduced closely to quantum limit.\n\n5.1.1 Theory of superconductivity\n\nIn 1911, Heike Kamerlingh Onnes (figure 5.1), a Dutch professor of experimental\nphysics, cooled mercury at the temperature of liquid helium to 4.2 K [89]. This\nexperiment does not only determine the boiling point temperature of Helium but\nalso measures the electrical resistance of the mercury. He discovered that the\nresistance of the metal dramatically drops to zero. The phenomenon called super-\nconductivity at low temperature and opened a new area in physics. In 1913, lead\nwas discovered as a superconductor at critical temperature Tc = 7.2 K, then 17\nyears after niobium was found as a superconductor at Tc = 9.2 K. Over the time\nmany metals and alloys were superconductors in between 1K and 18K.\n\nFigure 5.1: The liquid helium experiment of Heike Kamerlingh Onnes (on the\nright) and his chief technician Gerrit Flim. The historic superconducting plot\n\nof resistance versus temperature of the mercury at low temperature.\n\n\n\nTheory of a superconducting Transition-Edge Sensor 140\n\n1\n\n2\n\n3\n\n4\n\n5\n\n+\n\n- - -\n\nVibrations of lattice are minimal (virtual phonons) below Tc\n\nElectron traveling in front distorts the lattice\n\nVirtual phonons moved closer\n\n+\n\n+ +\n\n+ +\n\n+\n\n- - -\n+\n\n+\n+\n\n+ +\n+\n\nA positive region created behind the 1st electron\n\nThe 2nd electron is attracted to the positive region and  \nthe lattice rebounds back into its original shape\n\n+\n\n- - -\n+\n\n+ +\n\n+ +\n\nFigure 5.2: The BCS theory of superconductivity\n\nIn order to understand the quantum process of a superconductor, we need to\ninvestigate into the atomic study. In semiconductors or insulators, the energy gap\nof the order of\n\n3\n\n2\nkBT0 occurs between electrons and the periodic lattice. However,\n\nthe energy gap of superconductors mainly occurs due to phonon interaction of\nelectron and electron (≈ 100 nm), which is called Cooper-pairs [15]. The idea\nwas presented by John Bardeen, Leon Cooper, and Robert Schrieffer in 1957 and\nthey received the Nobel Prize in 1972 for the BCS theory of superconductivity.\nCooper-pairs are considered as Bosons (qq) and condense into the ground state.\nThe BCS microscopic theory of superconductivity was successfully described type\nI superconductor using quantum mechanics by the key idea Cooper-pairs. (1)\nWhen we cool down the sample into the critical temperature, the vibration of\nlattices are minimal (virtual phonon). (2) The electron traveling in front distort\nthe lattice. Therefore (3) the virtual phonon moved closer in lattice because of\nattracted negative first electron and thus (4) a positive region is created behind\nthe first electron. (5) The second electron has enough time to be attracted by the\nvirtual phonon before the lattice vibrated recoil to its shape, the pair of opposing\nspin electron is formed [6, 15]. The process is described in figure 5.2. In the end,\n\n\n\nPrinciple of a Transition-Edge Sensor (TES) 141\n\nthere is no resistance for Cooper-pairs, it means that electron does not slow down\nwith the lattice, then we have a perfect conductor or superconductor.\n\nIn simple words, the requiring energy to break the Cooper-pairs to normal elec-\ntrons (quasi-particles) is the energy gap of the superconducting state. At low\ntemperature, the zero resistivity of superconductor due to the thermal energy is\nless than the energy gap of the Cooper-pairs. The energy gap is decreased gradu-\nally to break the Cooper-pair when the temperature increased to the transition of\nthe critical temperature. The energy gap plays an important role in the theoretical\nexplanation of superconductivity and superconducting detectors. The BCS micro-\nscopic theory of superconductivity predicted that the energy gap is proportional\nto the critical temperature Tc:\n\nEg ≈\n7\n\n2\nkBTc ≈ 10−3eV. (5.1)\n\n5.1.2 The superconducting transition region\n\nIn solid state physics, a phase is defined by its structure, therefore, a phase transi-\ntion is a change of structure in geometry terms. Ehrenfest classified that there are\nonly two ways of phase transition first-order and second-order in thermodynamics\n[86]. The first-order phase is usually known as the transitions between different\ncrystal modifications involving latent heat absorption or relaxation, sudden vol-\nume change, discontinuous entropy, likely the water transforms into ice at the zero\ndegrees (from liquid to solid). Landau theory developed the second-order phase\nwhich specifically emphasized on the crystal symmetry changes continuously. The\nshape of the transition region depends on the material. The critical temperature\nof a thin film can be adjusted by its thickness because of exchanging Cooper-\npairs and quasiparticles or by using an alloy as the NbSi. The superconducting\ntransition region is extreme temperature sensitivity, a tiny changing temperature,\norder 0.1 mK to 1 mK leads to a large changing resistance. Therefore the su-\nperconducting transition edge sensor can be used as very sensitive detectors by\nbreaking Cooper-pairs which is order of meV compatible with mm detection which\nhas energy around E1mm = hc/λ1mm ∼ 1.2 meV.\n\n\n\nPrinciple of a Transition-Edge Sensor (TES) 142\n\n5.1.3 Principle of a Transition-Edge Sensor (TES)\n\nIn 1941, D.H. Andrews, American chemical professor, designed an experiment of\nan infrared source and a fine tantalum wire, which is operated in its superconduct-\ning phase transition at 3.2 K. He measured temperature versus resistance of the\ntantalum wire. The D.H. Andrews experiment was made the first Transition-Edge\nSensor (TES), also called a superconducting phase transition thermometer. The\nmain difficulty is to read out the low resistance of TES (few Ω). The supercon-\nducting quantum interference device (SQUID) with associated input loop is finally\nused to current read out a voltage biased TES [70].\n\n5.1.3.1 Electrical and thermal response\n\nFigure 5.3 illustrated a bolometer 1 and the electrothermal feedback playing an\nimportant role for TES. Bolometer converts energy to temperature and the natural\ntime constant is the ratio of the heat capacity and the thermal conductance. A TES\nis a bolometer exhibiting a strong electrothermal feedback. The electrothermal\nfeedback of the TES plays a key concept of the TES technology. The block diagram\nfigure 5.4 illustrated the feedback. The incident power is compensated by the Joule\ndissipation.\n\nPi\n\nPJ\n\nC , TcRTES\n\nG , Pbath\n\nTbath\n\n1\nG\n\n+\n[T ]\n\nElectric\n\nThermal\nP i\n\nP j\n\nP\n\nFigure 5.3: Left: The simple thermal bolometer model, Pi [W] , PJ [W] and\nPbath [W] are the incident signal power, the Joule power dissipated by the ther-\nmometer and the power to the bath temperature, respectively. C [J/K] is the\nheat capacity, G [W/K] is the thermal conductance, Tc [K] is the critical tem-\nperature of the TES, Tbath [K] is the reference bath temperature. Right: The\n\nelectrothermal feedback model.\n\n1A bolometer measures the power of incident radiation via the heating of a material.\n\n\n\nPrinciple of a Transition-Edge Sensor (TES) 143\n\n1\nG\n\n+\n[T ]\n\n−V\nR2\n\nV α\nR0\nT 0\n\n1\nG\n\nP i\n\nP j\n[T ][R][ A]\n\n+\n[ A]−V\n\nR2\n\nV\n\nα\nR0\nT 0\n\nP i\n\nP j\n\n[T ] [R]\n\n[ A]\n\n1\nG\n\nFigure 5.4: The left look likes a standard bolometer model: T(Pi). The\nright is a rearrange one highlighting the I(Pi) TES transfer function when it\nis voltage biased. R0, T0 are the steady state value of the TES resistance and\nTES temperature respectively. α is the logarithmic sensitivity to temperature\n\nparameter. R is the TES resistance under the voltage-biased operation.\n\nIbias\n\nRshunt\n\nIbias\nRpara\n\nMin\n\nRTES\n\nI\n\nVTH\n\nRL\nI\n\nMin\n\nRTES\n\nFigure 5.5: (left) The practical TES bias circuit and (right) its Thevenin\nequivalent circuit VTH = IbiasRshunt, the current is applied in the shunt resistor\nthen the load resistor in the Thevenin circuit RL = Rpara+Rshunt, the inductance\ncoil L with Min and the TES. The value of Rshunt and Rpara is of the order of 10\n\nmΩ.\n\nThe block diagram in figure 5.4 is useful to understand the TES response and\nthe mechanism of the negative electrothermal feedback effect which drives the\nthermometer back to the set point in its transition region. In the voltage-biased\n\nmode PJ =\nV2\n\nRTES\n\n, a rise in temperature of the thermometer leads to a rise in\nresistance, then the Joule power is decreased, it means that there is less power\nin the thermometer, thus it cools down because the Joule power compensated for\nthe original rising temperature. This process keeps a TES self-regulation in a set\npoint of the critical temperature.\n\nThe thermal and electrical response of the TES are described by circuits in figure\n5.5 and the figure 5.3 and follow two differential equations, which are the electrical\n\n\n\nPrinciple of a Transition-Edge Sensor (TES) 144\n\nequation for the current I though the TES, with the electrical resistance RTES.\n\nVTH = VTES + VL + VRL\n,\n\nL\ndI\n\ndt\n= VTH − IRL − IRTES. (5.2)\n\nand conventional energy, the thermal equation for the temperature T :\n\ndE\n\ndt\n= C\n\ndT\n\ndt\n= Pi + PJ − Pbath. (5.3)\n\nwhere I and T are the electrical current and the temperature of the TES, L is the\ninput inductance of the SQUID, VTH and RL is the Thevenin equivalent voltage\nand the resistance.\n\nThe electrical time constant of TES: τel\n\nFollowing the current conservation and the energy conservation, we can solve the\nThevenin circuit by understanding basic electric equations VR = IRL,VL = L\n\ndI\n\ndt\nin the biasing circuit figure 5.5. Then we develop the equation 5.2 by rearranging,\nintegration to find the solution.\n\ndI\n\nVTH − (RL + RTES) I\n=\n\ndt\n\nL\n,\n\nˆ\ndI\n\nVTH − (RL + RTES) I\n=\n\nˆ\ndt\n\nL\n,\n\n− ln (VTH − (RL + RTES) I)\n\nRL + RTES\n\n=\n1\n\nL\nt+ cst.\n\nSince I = 0 and t = 0, we have cst = − ln VTH\n\nRL + RTES\n\n, substituting back to the\nequation, we get:\n\nln (VTH − (RL + RTES) I)− ln VTH = −(RL + RTES)\n\nL\nt.\n\nTake exponentially to both sides then we have:\n\nI =\nVTH\n\nRL + RTES\n\n\n1− e\n\n−\nRL + RTES\n\nL\nt\n\n\n . (5.4)\n\n\n\nPrinciple of a Transition-Edge Sensor (TES) 145\n\nThe equation 5.4 indicates the transition period of current adjust from its initial\nvalue of zero to the final value\n\nVTH\n\nRL + RTES\n\n, which is the steady state. The intrinsic\ntime constant of the TES bias circuit is known as the electrical time constant:\n\nτel =\nL\n\nRL +RTES\n\n. (5.5)\n\nThe electrical time constant is the time for the current get to the final steady state.\nIndeed the inductor L creates a magnetic field in the circuit and store energy, below\nTc the resistance of the TES is zero therefore the time constant increases up to\nL\n\nRL\n\n.\n\nIn order to solve fully the electrical equation 5.2 and the thermal equation 5.3,\nwe will calculate independently each term of these equations such as the TES\nresistance RTES, the thermal bath power Ppath, the Joule power PJ. We will end\nup with TES’s time constants, the discussion of the electrothermal feedback effect,\nenergy resolution and TES responsivity.\n\nThe resistance of the TES: RTES\n\nWe can write the resistance of TESs as a function of temperature and current. This\nis called as non-linearity behavior of the thermometer resistance, RTES (T, I) =\n\nR0 + δR (T, I). R0, T0, I0 are steady state values, then we can expand a small\nsignal to the first order:\n\nRTES (T, I) ≈ R0 +\n∂R\n\n∂T\n\n∣∣∣∣∣\nI0\n\nδT +\n∂R\n\n∂I\n\n∣∣∣∣∣\nT0\n\nδI. (5.6)\n\nConveniently, we define two unitless sensitive parameters:\n\n\n\n\n\nα ≡ ∂ log R\n\n∂ log T\n\n∣∣∣∣∣\nI0\n\n=\nT0\n\nR0\n\n∂R\n\n∂T\n\n∣∣∣∣∣\nI0\n\n, the logarithmic sensitivity to temperature.\n\nβ ≡ ∂ log R\n\n∂ log I\n\n∣∣∣∣∣\nT0\n\n=\nT0\n\nR0\n\n∂R\n\n∂I\n\n∣∣∣∣∣\nT0\n\n, the logarithmic sensitivity to current.\n\n(5.7)\n\nFor a semiconductor the value of α is around -1 to -10, in contrast, it is 100 - 1000\nfor a superconductor, the electrothermal feedback is strong when α is high. We\ncan see the value of α for several TESs in the array P57 of the QUBIC experiment\n\n\n\nPrinciple of a Transition-Edge Sensor (TES) 146\n\nin figure 5.6. The resistance of the TES now is:\n\nRTES (T, I) ≈ R0 + α\nR0\n\nT0\n\nδT + β\nR0\n\nI0\n\nδI. (5.8)\n\nFrom the equation 5.8 we can derive the dynamic resistance of the TES is\n\nRdyn =\n∂V\n\n∂I\n\n∣∣∣∣∣\nT0\n\n=\n∂ [R (T0, I) I]\n\n∂I\n=\n\n∂\n\n[(\nR0 + β\n\nR0\n\nI0\n\n)\n(I0 + δI)\n\n]\n\n∂I\n,\n\n=\n\n∂\n\n(\nR0I0 + βR0δI + R0δI + β\n\nR0\n\nI0\n\nδI2\n\n)\n\n∂I\n.\n\nThe R0I0 is a constant, we can neglect the second order of the current term β\nR0\n\nI0\n\nδI2,\nthen the dynamic resistance of the TES is approximately expressed as:\n\nRdyn ≈ βR0 + R0 = R0 (1 + β) . (5.9)\n\nThe thermal bath power: Pbath\n\nThe thermal response of the TES around steady-state (R0, T0, I0) values can be\nderived from the thermal equation 5.3. Firstly we have the thermal conduction is\ndefined:\n\nG (T) =\n∂Pbath\n\n∂T\n= nκTn−1. (5.10)\n\nThe heat bath is assumed as an exponential function:\n\nPbath =\n\nˆ T0\n\nTbath\n\nG (T) = κ (Tn\n0 − Tn\n\nbath) . (5.11)\n\nIn the equilibrium state T0, the rising temperature of the absorber T0 + δT flows\nto the thermal conduction of the thermal link to the bath temperature.\n\nPbath\n\n∣∣∣∣∣\nT0\n\n≈\nˆ T0+δT\n\nTbath\n\n≈ G (T) dT,\n\n≈ Pbath0 +\n∂Pbath\n\n∂T\n\n∣∣∣∣∣\nT0\n\n,\n\n≈ Pbath0 + GδT,\n\n≈ κ (Tn\n0 − Tn\n\nbath) + nκTn−1\n0 δT. (5.12)\n\n\n\nPrinciple of a Transition-Edge Sensor (TES) 147\n\nFigure 5.6: The plot of temperature versus resistance and the α parameter for\npixel 1, 2, 3, 4 of the 256 TES array named P57. [Damien Prêle-APC/Stefanos\n\nMarnieros-CSNSM, personal communication, 2018 ]\n\nwhere δT = T − T0, κ, and n depends on the natural thermal link and the heat\nbath temperature. In practice, for the TES the value of n ∼ 3 or 4. κ depends\non the type of using materials of pixel’s legs, in this case of the TES, we are using\nSiN. The equilibrium state of the bath temperature is Pbath0 = PJ0 + Pi0 . Pi0 is\nthe equilibrium state of the incident signal and the equilibrium state of the Joule\npower is PJ0 = I2\n\n0R0.\n\nThe Joule power: PJ\n\nIn the current biased operating mode, the Joule power is calculated as PJ =\n\nRTES (T, I)× I2\nbias while in the voltage biased operating mode, the first order of the\n\n\n\nPrinciple of a Transition-Edge Sensor (TES) 148\n\nJoule power can be calculated around (R0,T0, I0).\n\nPJ =\nV2\n\nTES\n\nRTES (T, I)\n= I2RTES (T, I) = (I0 + δI)2 RTES (T, I) ,\n\n≈ PJ0 + 2I0R0δI + α\nPJ0\n\nT0\n\nδT + β\nPJ0\n\nI0\n\nδI. (5.13)\n\nWe will discuss after that the TES is stable in the voltage biased operating mode\nvia the presence of the strong negative electrothermal feedback effect. We define\nthe low-frequency loop gain constant in the voltage-biased operation and natural\ntime constant of thermal response [61].\n\nL ≡ PJ0α\n\nGT0\n\n=\nI2\n0R0α\n\nGT0\n\n,\n\nτ ≡ C\n\nG\n. (5.14)\n\nThe thermal time constant: τth\n\nWe substitute the TES resistance equation 5.8, the bath power equation 5.12, the\nJoule power equation 5.13 and the low frequency loop gain equation 5.14 into the\nthermal and the electrical different equations 5.3, 5.2. In addition, we substitute\nalso the variables T − T0 = δT, I − I0 = δI and we just only consider the first\norder of those equations, finally we got new equations [61]\n\n\n\n\n\ndδI\n\ndt\n= −RL + R0 (1 + β)\n\nL\nδI− L G\n\nI0L\nδT +\n\nδVTH\n\nL\n.\n\ndδT\n\ndt\n=\n\nI0R0 (2 + β)\n\nC\nδI− 1−L\n\nτ\nδT +\n\nδPi\n\nC\n.\n\n(5.15)\n\nWhere δVTH ≡ Vbias − V0 is the changing voltage bias around the equilibrium\nstate, δPi ≡ Pi − Pi0 is the changing signal around the equilibrium state. We can\nrecognize that the electrical time constant of TES is described in the equation 5.4\n\nτel =\nL\n\nRL + R0 (1 + β)\n=\n\nL\n\nRL + Rdyn\n\n. (5.16)\n\n\n\nPrinciple of a Transition-Edge Sensor (TES) 149\n\nand the thermal time constant is defined by the nature time constant of the TES\nand the low-frequency loop gain constant:\n\nτth ≡ τeff ≡ τ1 =\nC\n\nG(1 + L )\n=\n\nτ\n\n1 + L\n. (5.17)\n\nNotice here that L is the negative loop gain feedback in a voltage biased oper-\nating mode, it is equivalent to LV and LV = −LI due to the α the logarithmic\nsensitivity temperature parameter. The equation 5.17 proved the advantage of a\nTES, the |L | > 1 then the natural time constant of a bolometer τ is decreased\nby the effective loop gain of the electrothermal feedback effect and thus the speed\nof a TES is increased.\n\nSo far we have demonstrated the electrical and thermal responding time constants\nof a TES thought TES differential equations. The 2 time constants and the feed-\nback imply that the response of the TES can be stable or unstable.\n\nTES responsivity: s(ω)\n\nWe can rearrange equations 5.15 in the matrix form.\n\nd\n\ndt\n\n\n\nδI (t)\n\nδT (t)\n\n\n = −\n\n\n\n\n1\n\nτel\n\nLG\n\nI0L\n\nI0R0 (2 + β)\n\nC\n\n1−L\n\nτ\n\n\n\n\n\n\nδI (t)\n\nδT (t)\n\n\n+\n\n\n\n\nδVTH\n\nL\n\nδPi\n\nC\n\n\n . (5.18)\n\nThe solution of electrical and thermal response equations 5.18 are exponential\nfunction by taking the non-homogeneous term δVTH, δPi to zero [61]. After calcu-\nlating eignvalues and rewriting in term of the inductance, we found that the power\nto the current responsivity is expressed by [61]:\n\ns(ω) = − 1\n\nI0R0\n\n[\nL\n\nτelR0L\n+\n\n(\n1− RL\n\nR0\n\n)\n+ jω\n\nLτ\n\nR0L\n\n(\n1−L\n\nτ\n+\n\n1\n\nτel\n\n)\n− ω2τL\n\nL R0\n\n]−1\n\n.\n\n(5.19)\nThe equation 5.19 is the TES responsivity related to the loop gain L . The TES\nmodel in figure 5.3 also implies that a fluctuation of the incident power to the TES\nis transformed to the resistance of the thermometer by a fluctuation in the Joule\npower.\n\nL =\n∂T\n\n∂P\n\n∂PJ\n\n∂T\n=\n∂T\n\n∂P\n\n∂PJ\n\n∂R\n\n∂R\n\n∂T\n. (5.20)\n\n\n\nPrinciple of a Transition-Edge Sensor (TES) 150\n\nFigure 5.7: The linear response of the current versus the heat power [69].\n\n∂T\n\n∂P\nis the expected transfer function of a bolometer from power to temperature.\n\nThe signal depends on the fluctuation of the TES resistance\n∂R\n\n∂T\n, the sensitive\n\ntemperature parameter α and the operating mode of voltage-bias or current-bias.\n∂PJ\n\n∂R\nis the transfer function of the electrical feedback system, it depends on the\n\noperating mode of voltage-bias or current-bias.\n\nIn the voltage-biased mode of the supercondicting TES, the strong negative elec-\ntrothermal feedback reduces the nature time constant of a TES up to two orders of\nmagnitude which is necessary to produce a high resolution signal, and the current\nresponsivity to the absorbed power is proportional to the inverse of the voltage\nbias. From the equation 5.19, the current responsivity of the superconducting TES\nis given [69, 70, 125]:\n\nsI (ω) = − 1\n\nVbias\n\n|L |\n1 + |L |\n\n1\n\n1 + jωτeff\n\n[A/W]. (5.21)\n\nFigure 5.7 presents the historic experiment which demonstrates the voltage-biased\nsuperconducting TES has a linear response of the current to the heat power fol-\nlowing the\n\n1\n\nVbias\n\n[69].\n\nThe electrothermal feedback (ETF) effect\n\nThe electrothermal feedback (ETF) effect is the interaction between the electric\ncurrent and the thermal temperature through the change in the resistance of the\nTES. It has shown in the cross terms of the electrical and thermal equations 5.18.\nThe value of the α parameter of the transition edge sensor is positive as shown\nin figure 5.6, the increasing temperature will increase the electrical resistance of\nthe TES and the electric current in the TES dissipates the thermal power to the\n\n\n\nPrinciple of a Transition-Edge Sensor (TES) 151\n\nresistance of the TES by Joule power. In case of current-bias ( RL � RTES), the\nJoule power is calculated by the formula PJ = I2RTES =⇒ the ETF is positive,\nit means unstable. In the case of voltage-bias (RTES � RL), the Joule power is\n\ncalculated by the formula PJ =\nV2\n\nRTES\n\n=⇒ the ETF is negative, it means stable.\nThere has no stable margin in case of RL = RTES. Advantages of the current-\nbias are that the voltage can be easily amplified. Nevertheless, the TES become\nunstable. Furthermore, we implement the Superconducting QUantum Interference\nDevices (SQUID) which makes possible to do voltage-bias [70]. In the voltage-\n\nbiased mode, the Joule power is given by PJ =\nV2\n\nRTES\n\nthen an increasing incoming\n\nsignal (it means also increase of T and RTES) will decrease the Joule power, then\nthe total power on the TES will be staying to a constant. This is called the\nnegative ETF effect:\n\n• A strong negative ETF speeds up the detector, the thermal time constant\nof the effective thermal time constant of a TES is divided by the loop gain\nL , the result is demonstrated by theory and experiment.\n\n• In the voltage-biased mode, the TES is self-calibrating staying within its\ntransition temperature, this physical characteristic is useful for an array of\nTESs and is one of the main interest of the use of TES instead of other kinds\nof bolometer.\n\n• The current response is − 1\n\nVbias\n\n.\n\n5.1.3.2 Noise performance\n\nNoise is unwanted and random signals, the main types of noise are thermodynamic\nnoise, low frequency (1/f) noise and shot noise. The thermal noise is observed by\nJ. B. Johnson in the Bell Telephone Laboratory in 1927 and studied by H. Nyquist\nin 1928 [2]. The electrons in a conductor are random vibrations in the lattice and\ndepend on the temperature thus there have many tiny currents inside a conductor\nbecause an electron has charged 1.6× 10−19 Coulomb. The thermal noise power is\nbasically calculated by the temperature and a bandwidth of a measurement ∆f .\n\nP = kBT∆f. (5.22)\n\n\n\nPrinciple of a Transition-Edge Sensor (TES) 152\n\nwhere T is the temperature of the conductor in Kelvin, kB is the Boltzmann’s\nconstant. In the room temperature, for 1 Hz bandwidth, the noise power is in\norder of 4× 10−21W . It is equivalent to - 204 dBW [2].\n\nThe sensitivity of an operating TES is determined by different noise sources:\n\n• The thermal fluctuation noise (TFN) or phonon noise dues to the link be-\ntween the intrinsic detector and the bath.\n\n• The Johnson or Nyquist noise which is associated with the resistors of the\nTES circuit.\n\n• The noise from the different stage of the electronic readout system as SQUID\nnoise, ASIC noise . . .\n\nNoise Equivalent Power (NEP)2 is understood as the harmonic input noiseW/\n√\nHz.\n\nThe low value of NEP means the highly sensitive detector. In general, we can ex-\npress the NEP through the power spectral density (PSD), which characterizes a\nstationary random process by taking the square of the signal, in the frequency\ndomain the PSD of the signal x (t) is calculated:\n\nPSDx(ω) =\n1\n\n2T\n\nˆ T\n\n−T\n\n∣∣∣∣∣x (t) e−j2πωt\n\n∣∣∣∣∣\n\n2\n\ndt. (5.23)\n\nBecause a stationary stochastic process is typically not an absolute integration\nthen the signal has to be truncated before the Fourier transform. The relationship\nof the NEP and the PSD is given by:\n\nNEP =\n√\n\nPSD\n[\nW/\n√\n\nHz\n]\n. (5.24)\n\nIn case of noise sources are not correlated, the total NEPtot can be understood as\nvariations and we can sum separately:\n\nNEPtot2 = NEP2\nhν + NEP2\n\nint + NEP2\nelec.\n\nNEP2\nint = NEP2\n\nphonon + NEP2\nJRL\n\n+ NEP2\nJTES\n\n. (5.25)\n\n2Noise Equivalent Power (NEP) is defined as the ratio of an input signal and the electrical output\nsignal given by W/\n\n√\nHz or in simple words, the NEP is the necessary power to equalizing noise\n\nlevel during a second.\n\n\n\nPrinciple of a Transition-Edge Sensor (TES) 153\n\nwhere NEPhν and NEPelec are the photons noise and the electronic system noise,\nrespectively. NEPint is the intrinsic noise of TES components.\n\nPhotons noise\n\nwhen an experiment observes the CMB photons by its focal plane, obtaining pho-\ntons in a TES follow a Poisson distribution. The measurement of photons gives\nintrinsic photons noise. An average number of photons 〈n〉 (bosons) obey Bose-\nEinstein statistic.\n\n〈n〉 =\n\n\ne\n\nhν\n\nkBT − 1\n\n\n\n\n−1\n\n. (5.26)\n\nwith ν is the frequency of electromagnetic waves and T is the source temperature.\nThe variance of number of photons is given\n\n〈∆n2〉 = 〈n〉+ 〈n〉2. (5.27)\n\nThe power of the photons is multiplication of a single photon energy and the\naverage number of photons considering the efficient responding frequency η (ν):\n\nPγ =\n\nˆ\nν\n\nhν〈n〉 η (ν) dν =\n\nˆ\nν\n\nhν\n\n\ne\n\nhν\n\nkBT − 1\n\n\n\n\n−1\n\nη (ν) dν. (5.28)\n\nApply the equation 5.27 we have the variance of power formula\n\n(\n∆P2\n\n)\n=\n\nˆ\nν\n\n(hν)2 η (ν) 〈n〉 (1 + η (ν) 〈n〉) dν. (5.29)\n\nThe variance power is calculated in a second which is corresponded 0.5 Hz band-\nwidth in the Fourier domain so that the integration is from 0 to 1/2. Then the\npower spectral density is square of the NEP over 2 [76].\n\nNEP2\nγ\n\n2\n=\n(\n∆P2\n\n)\n=\n\nˆ\nν\n\n(hν)2 η (ν) 〈n〉 (1 + η (ν) 〈n〉) dν. (5.30)\n\nIn case of a square function or a box car filter the η (ν) = 1. and the bandwidth\nis less than the frequency center ∆ν � ν0.\n\nNEPγ ≈\n\n√\n\n2hν0Pγ + 2\nP2\nγ\n\n∆ν\n. (5.31)\n\n\n\nTES arrays of the QUBIC experiment 154\n\nwith Pγ = hν0〈n0〉∆ν. When hν � kBT photons noise is called Bose bunching\nnoise, and hν � kBT is called Poisson noise.\n\nThermal fluctuation noise or phonon noise\n\nThe exchange energy of a detector (thermometer) with the thermal bath link at\nlow-temperature dues to processes of phonons or quasi-particles under the quan-\ntum mechanic physics which are vibrations of atoms in the lattice because fluc-\ntuations in the temperature lead to fluctuations in the resistance as well as the\nelectrical current. In the TES the approximation of the NEP for the phonon noise\nat the temperature T with the thermal conduction G is given by the formula [77]\n\nNEPTFN =\n√\n\n4kBT2G\n[\nW2/Hz\n\n]\n. (5.32)\n\nJohnson noise of the thermometer resistance (RTES) and the load resis-\n\ntance (RL)\n\nThe ETF reduces the TES Johnson noise by the Loop gain of the TES. Then the\npower spectral density of the current noise due to the Johnson noise of the resistor\nof the TES thermometer is given [61]:\n\nPSDITES\n≈ 4kBT0/R0/L\n\n2, (5.33)\n\nfor the shunt resistor, the noise is not affected by the feedback of Loop gain of the\nTES and stay:\n\nPSDIshunt ≈ 4kBTL/Rshunt (5.34)\n\nElectronic gain\n\nThe SQUID is the first stage of readout the TES signal, then the noise of the\nSQUID amplifier noise is encountered with the gain of the amplifier.\n\nGelec =\nMin\n\nMfb\n\nRfb. (5.35)\n\nWhere Min,Mfb are transformer coils of a SQUID and the flux-locked loop. Rfb is\nthe feedback resistor of the flux-locked loop.\n\n\n\nTES arrays of the QUBIC experiment 155\n\n5.2 TES arrays of the QUBIC experiment\n\nThe QUBIC’s TES array is an array of 256 TES pixels. Each pixel has 30 nm\nthickness of superconductor NbSi thin film with 15.45 % of niobium3, absorbing\ngrids are made of TiV. The sheet resistance of the TiV thin film is matched\n(increases) to the vacuum impedance Z0 by shaping the absorber as a grid. The\nsheet resistance of the absorber is then given by equation 5.36\n\nRS =\nρ\n\nt\n× L\n\nl\n. (5.36)\n\nWhere ρ is the resistivity Ωm of the TiV, t is the thickness and L, l the geometry\nparameters of the grid as shown in figure 5.8. The critical temperature of the TES\n\nFigure 5.8: Geometry parameters of the absorbing grid\n\ndepends on the percentage components of the niobium and silicon as shown in\nfigure 5.9. The ratio of niobium and the thickness of the film decides to the goal\ncritical temperature thus the requiring critical temperature of the QUBIC’s TES\narray is Tc ≈ 400 mK [85].\n\nTo build a TES array, the first point is that we have to define the dimension of a\npixel as well as the architecture of the array that has to fit with the focal plane of\nthe QUBIC experiment at 150 GHz (λ = 2 mm) and 220 GHz (λ = 1.4 mm). Due\nto the incident photons, we want to measure, the absorbing grid surface of the\nQUBIC’s TES is required ≥ 2 mm. The resolution of the instrument relates to the\n3Niobium (Nb) has atomic number 41.\n\n\n\nTES arrays of the QUBIC experiment 156\n\nFigure 5.9: The critical temperature of a TES depending on the percentage of\nniobium. Two techniques have been presented in the plot, co-evaporation (dot\n\nline) and co-sputtering (bar line) [94].\n\nlimited diffraction for a collected photons mirror (diameter D) which is express:\n\nθ = 1.22\nλ\n\nD\n. (5.37)\n\nThe QUBIC telescope has a primary mirror diameter of about D = 317 mm then\nwe have the table of angular resolution.\n\nTable 5.1: The table of angular resolution\n\n150 GHz D = 317 mm θ = 26.5 arcminus\n220 GHz D = 317 mm θ = 18.5 arcminus\n\nThe TES array is fabricated in the clean room using microfabrication technologies\nwhich are performed in the Center for Nanoscience and Nanotechnology (C2N)4\n\nlaboratory and the Centre de Sciences Nucléaires et de Sciences de la Matière\n(CSNSM). The cross section of the successive processes to build one pixel is shown\nin figure 5.11. Microfabrication is a sequential multi-processes technology , its com-\nplexity is described by mask count layers. The mask of the 256 TESs is shown in\nfigure 5.10. We use 500 µm thick of a silicon on insulator (SOI) substrate with 3\ninches (∼ 7.6 cm) perimeter, the substrate is two layers of silicon and a thin layer of\n4C2N was established in 2016 by the joint of CNRS and University Paris Sub laboratories on the\nsame campus: Laboratory for Photonics and Nanostructures (LPN) and Institut d’Electronique\nFondamentale (LEF).\n\n\n\nTES arrays of the QUBIC experiment 157\n\nFigure 5.10: The mask of 256 TES pixels array, the magenta curve is 3 inch\n( 7.6 cm) wafer perimeter. Each TES has Al wires connection to the bonding\n\nwire.\n\nsilicon dioxide SiO2 5 µm. The SiO2 layer not only helps to stop the deep etching\nun-uniform between different areas on the array but also helps to produce a homo-\ngeneous pixel array. The membrane layer is 1 µm thick silicon nitride Si3N4 and\nthen a superconducting NbSi-thermometer and aluminum-wires are evaporated in\norder on the top of the membrane layer. The next step, the metallic absorbing\ngrid (TiV or Pd) is added on the top and releasing the membrane. Finally, the\nlegs are excavated [93]. Figure 5.11 illustrates the microfabrication performance\nof a pixel. There are 12 steps in the process, the detail of the microfabricating\nperformance is given in the Camille Perbost thesis (in French) and her team in\nCSNSM, C2N and APC laboratories [94].\n\nThe Si3N4 membrane has high strength and has been widely used in micromachin-\ning, then legs and absorber structures can be fabricated. The thermal properties\nof this membrane at a low temperature allowing the mean free path of phonons\nis long then the heat from the absorber can transform immediately to the super-\nconducting thermometer. The thermal conduction of the Si3N4 is also limited by\n\n\n\nTES arrays of the QUBIC experiment 158\n\nFigure 5.11: TES fabrication is used the microelectromechanical systems\n(MEMS). The processes are preparing substrate, evaporated membrane, added\nthermometer and wire, added absorbing grid. credit: Camille Perbost/Christelle\n\nCarré [94].\n\n\n\nTES arrays of the QUBIC experiment 159\n\nFigure 5.12: From the top to the bottom, the left to the right: A single pixel\ncomponents, the substrate is layers of Si, SiO2, the pixel is layers of Si3N4, Al,\nTiV and NbSi. The dimension of a pixel is ≈ 3 mm. The microphotography of\na superconducting NbSi thermometer which is located at the center of the pixel.\n\nThe 256 TESs array (named P90) and the copper backshort.\n\nthe scattering of phonons in the surface, therefore the low thermal conductivity is\nmaintained. The silicon-on-insulator wafer has two layers of Si which are separated\nby an amorphous SiO2, this technology allows reach to low thermal conductivity\nG structures.\n\nFigure 5.12 shows the dimension of a pixel as well as a zooming of the super-\nconducting NbSi thermometer at the center of a pixel. This figure also presents a\nfabricated array P90 and its bonding wires with the copper backshort which allows\nmounting on the focal plane of a cryostat. These superconducting wires will be\nconnected to the SQUIDs.\n\nAfter the successful fabrication of a TES array which occurs 10 % of the case,\nwe need to do some tests. (1) The first test is the connection of Al wires and\nTES using measurements of TES resistances at room temperature. We applied\na DC bias measuring current from -0.1V to 0.1V (step 0.02V) to the aluminum\n\n\n\nThe cryostat and the electronic readout system 160\n\nwires which always connect with the superconducting NbSi thermometers. The\nresistance of aluminum wires and the thermometer metal is deduced I =\n\nV\n\nR\n. Figure\n\n5.13 presents the cartography and histogram of the TESs resistance of the array\nat room temperature measurement. We can approximately an Al wire ( ∼ 10 cm\nlength, ∼ 200 nm thickness, ∼ 6 µm width). Therefore the value of the TESs\nresistance at the room temperature test is of the order of 2 kΩ. Due to the fact\nthat we carry out measurements of the superconducting NbSi thermometer plus\nthe Al wires thus the values of TES resistance of the array P90 in figure 5.13 are of\nthe order of ∼ 4000 Ω compatibly cause aluminum resistance wires. The values of\nTES resistance have varied from different arrays. The goal of this measurements\nis the test of connections between TESs and Al wires and through it evaluating\nthe yield of a TES array. The cartography demonstrated a yield of the TES array\nabout 85 % at room temperature test. Due to the SQUID and wire bonding issue,\n10 % more of TESs will be lost, finally, the yield of the TES array inside the\ncryostat is ≈ 75 % after the electronic readout system. This measurement helps\nprimarily to know the yield as well as resistors distribution of the array, therefore\nwe can decide the array is useful or useless. Table 5.2 presents array names and its\nyield of good pixels. Some arrays perform a good yield number in the connection\ntest in the room temperature, however, there have some reasons such as the values\nof TES resistance are too high or un-uniform, many broken pixels or problems of\nthe silicon wafer . . .We base on that measurements and reasons to evaluate and\ndecide a TES array. (2) The next step, the good TES arrays will be bonded and\nfixed in a copper backshort as shown in figure 5.12, which can be plugged with the\nfocal plane and keep the array safe in movements. Then the TES array is installed\ninto the focal plane inside the cryostat, which is cooled down to 300 mK. The TES\nresistance, the I-V curves, radioactive source test will be tested and readout by\nthe electronic system.\n\n5.3 The cryostat and the electronic readout sys-\n\ntem\n\nThe test cryostat in the APC laboratory is a component of a Triton 200/400 sys-\ntem. The Triton system has 6 connected independent components: the cryostat\ncooled system, the control box, the power supply, the pump system, the compres-\nsor, and the liquid nitrogen. The detail of the Triton system is described in the\n\n\n\nThe cryostat and the electronic readout system 161\n\n0 5 10 15\n\n0\n2\n4\n6\n8\n\n10\n12\n14\n16\n\nCartographie ( )\n\n0\n\n2000\n\n4000\n\n6000\n\n8000\n\n10000\n\n0 5000 10000 15000\nResistance  \n\n0\n\n10\n\n20\n\n30\n\n40\n\nNo\n o\n\nf T\nES\n\ns\n\nTES array\n\nFigure 5.13: The measured resistor result of connection between NbSi and\nAl wires for the array P90 at room temperature. The cartography of TESs\nresistance indicates that there are 15 % of dead pixels which are in the black\ncolor. The histogram of TES resistance for good pixels, these values of resistance\n\nare of the order of 4000 Ω.\n\n\n\nThe cryostat and the electronic readout system 162\n\nNumber Array name Yield (good pixels) Evaluation & Current status\n1 P63 84 % Good & Usable\n2 P65 84 % Ok & Usable\n3 P68 77 % Ok & Usable\n4 P71 75 % Not good & Useless\n5 P73 89 % Good & Usable\n6 P82 77 % Not good & Useless\n7 P86 92 % Ok & Usable\n8 P88 84 % Ok & Usable\n9 P90 88 % Good & Usable\n\nTable 5.2: The table of TES arrays.\n\ntechnical manual [3]. Figure 5.14 describes different temperature stages of the\ncryostat, a TES array is located and operated in superconducting temperature at\nthe mixing chamber (MC plate).\n\nIn the vacuum tube, the cryostat system cools the temperature of the mixing\nchamber down to the temperature (mK) base on the helium dilution refrigerator.\nBasically, the principle is that a mixture of 3He+4 He is in equilibrium with 3He\n\nin the mixing chamber. When 4He is added to the mixture, 3He is evaporated,\nthis diluting process is an endothermic process, thus the temperature of the system\nis cooled down by the absorbed heat process. The system has the pump loop to\npump the 3He back the mixture, then the temperature of the mixing chamber\ncontinues to cool down to the setpoint of the controlled system.\n\nThe TESs array is mounted in the focal plane and the wire is connected to 256\nSQUIDs multiplexer, the ASIC, the warm digital readout FPGA board which\nhas included PID controller and the scheme of a TES array is showed on the\nscreen computer by the QUBIC studio software. Figure 5.15 shows the order of\nthe electronic readout system devices while Figure 5.16 illustrates its equivalent\nscheme. The focal plane of the QUBIC experiment is kilo-pixel TESs composed of\n4 quadrants of 256 TESs, each read out by 256 SQUIDs and 2 ASICs. The readout\nof kilo-pixel TESs requires a high and complex technology readout system as the\ncryogenics system, the ultra-low noise cryogenic amplification, the flux locked loop,\nADC, FPGA.\n\nDescriptions of readout components:\n\n• A 128-to-1 Time-Domain SQUIDMultiplexer: Superconducting QUan-\ntum Interference Device (2 mm x 3 mm) is used to readout voltage-bias\n\n\n\nThe cryostat and the electronic readout system 163\n\nFigure 5.14: The vacuum chamber of Triton 200/400, dilution refrigerator is\nthe stage of under 3.5 K. OVC, PTR stand for Outer Vacuum Chamber and\nPulse Tube Refrigerator, respectively. PT1, PT2 are Pulse Tube first, second\nstage. IAP is Intermediate Anchoring Plate (100 mK plate). MC plate stands\n\nfor the mixing chamber plate [3].\n\nTESs, SQUIDs are operated around 0.1-4 K. Due to the fact that the yield of\nSQUIDs is not 100 %, then independently a TES is read out by a SQUID. A\n128-to-1 Time-Domain SQUID Multiplexer has been investigated and man-\nufactured for the QUBIC experiment to readout kilo-TESs [123]. Therefore\neach quadrant focal plane has a 256 TESs array which is composed of 2\nTime-domains SQUID multiplexers. 128 SQUIDs are arranged in 4 columns\nand 32 rows as shown in figure 5.17.\n\nThe second multiplexing stage acts as 128-to-4 multiplexer, the ASIC and\nthe low noise amplifier (LNA) at 40 K stage read out the SQUIDs as a 4-to-1\nmux. A SQUID adds about 10 pW per TES and this power is dissipated\n\n\n\nThe cryostat and the electronic readout system 164\n\nTES SQUID ASIC FPGA\n\nCryostat wiring for 256 TESs warm \nreadout FPGA & QUBIC studio\n\nC\nry\n\nos\nta\n\nt\n\n256xTES  @T=300mK\n\n128:1xSQUID  @T=1K\n\nx4\n\nASIC @T=40K\n\nInterconnection PCB @T=4K \n\nC\nKb\n\n\n\n(M\n\nU\nX)\n\nSY\nN\n\nC\nC\n\nb\n\nSY\nN\n\nC\nLb\n\nIS\nQ\n\n_R\nEF\n\nVD\nD/\n\nG\nN\n\nD\nAn\n\nal\nog\n\n \n\nR_\nsh\n\nun\nt\n\n\n(T\nES\n\n b\nia\n\ns)\nR_\n\nfb\n S\n\nQ\nU\n\nID\n\n\n\n(F\nLL\n\n)\nSQ\n\nU\nID\n\n \n\nhe\n\nat\ner\n\nFigure 5.15: Readout electronic components of the cryostat in the APC lab-\n\noratory in order of connection the TES array is mounted with the\n1\n\n4\nQUBIC\n\nfocal plane, the SQUID, the ASIC, the warm readout FPGA board.\n\nIsq\nLNA ADC\n\nDAC compensation\n\nerror signal\n\nRfb\n\nφfb\n\nφinTESIbias\n\nRsh\n\nTES bias SQUID ASIC Warm Digital Readout\n\nFigure 5.16: The scheme of the TES readout system from sub Kelvin to warm\ndigital readout.\n\n\n\nThe cryostat and the electronic readout system 165\n\nFigure 5.17: The topology of the 128:1 Time-Domain SQUID multiplexer.\nEach column has 32 rows of SQUIDs. 4 columns are read out by ASIC and a\nlow noise amplifier (LNA). Capacitors Cbias allows to isolate the voltage of each\n\nSQUID [123].\n\nto the cryostat shield. The sample rate of the readout 128 TESs in the\ntime domain is shown in figure 5.18. The first cycle sample is \"Mux\" of\nthe 1st pixel to the 128th pixel then return to the 1st pixel. A frequency\nsample fs is chosen by \"QUBIC studio\" then we can calculate the \"Line\",\nit is corresponding to the sample rate of each pixel or inverting of frequency\nacquisition facq. There has mismatch step between each pixel due to the\nconstraint of the dynamic readout range, in practice with \"QUBIC studio\",\nwe will not take data of some beginning points of \"Mux\".\n\nsample rate =\n1\n\nfacq\n\n=\n1\n\nfs\nNdetsNpts [s]. (5.38)\n\nwhere facq is the frequency acquisition, Ndets is number of pixels, Npts is\nnumber of measurements done for each pixels. For example, we read 128\nTESs with frequency 2 MHz for 1000 samples per pixel, then we got the\nsample rate of each pixel is 0.064 s.\n\n• The ASIC: Application Specific Integrated Circuit is designed to read out\nSQUID/TES at cryogenic temperature, in addition, Low Noise Amplifica-\ntion (LNA). The electronic circuit of the resistor and heterojunction bipolar\n\n\n\nThe cryostat and the electronic readout system 166\n\nFigure 5.18: The sampling rate of 128 TESs, the Mux read out 128 TESs,\nCycle and Line signal provided by ASIC [123].\n\ntransistors, on a few mm2 chip is needed to fit with the intrinsic voltage\nSQUID noise (nV/\n\n√\nHz). The functions of LNA are used to provide a large\n\ngain bandwidth ( � 1 GHz), small 1/f noise, basically, the noise is affected\nby a factor of square root of the number of detectors\n\n√\nNdets because of the\n\nmux. The ASIC of the QUBIC experiment uses commercial of BiCMOS\nSiGe 0.35 µm technology, and operates around 2 - 300 K. The ASIC ampli-\nfies and reads the signal from the 4 SQUID columns of 32 SQUIDs through 4\nmultiplexed inputs of the LNA. Sequentially, the analog signal is a timeline\nof 128 TESs through SQUIDs and LNAs [124].\n\n• The FPGA is the abbreviation for Field Programmable Gate Array, which\nis programmed by Very high speed integrated circuit Hardware Description\nLanguage (VHDL) language. FPGA is a warn electronic readout at the room\ntemperature. We are using FPGA XEM3005 Xilinx Spartan-3E, 32 MB, 16-\nbit wide SDRAM, USB connection, fast configuration, easy communication,\nand access. The purposes of the FPGA board are compensated offset to\ncharacterize the SQUID signal, controlled and feedback the flux locked loop\nof the system by loading PID controller parameters from the user (developed\nby the Research Institute in Astrophysics and Planetology (IRAP)).\n\n• The PID controller: PID stands for Proportional Integral Derivative. PID\ncontroller is widely used in an automation system. In this system it controls\na loop feedback. The PID controller is programmed and embedded inside the\nFPGA board. P is using to control compensated error between the setpoint\nvalue and the real value. I learn from past values of the system and control\nto eliminate the residual error. D estimates the future trend of the setpoint.\n\n• The QUBIC studio is a software tool. It is installed on a window com-\nputer. The QUBIC studio can perform TES readout signal by an intuitive\n\n\n\nThe cryostat and the electronic readout system 167\n\nFigure 5.19: The QUBIC studio interfaces on a window operation computer.\n\ncartography on a computer screen, saved data, compensated offset, voltage-\nbiased TESs. The QUBIC studio helps to calibrate the TES array, measure\nI-V curves. The interface of the QUBIC studio is shown in figure 5.19.\nThe software is integrated a pystydio and a qubic pack. They are python\ninterfaces control locally or remotely read and write, data analysis, data\nplotting. The pystudio and the qubic pack are available on the GitHub:\nhttps://github.com/satorchi/pystudio\n\nThe TES and the readout system can be transformed to block diagrams as shown\nin figure 5.20. This algorithm can be used to express the electrical time constant\nof the TES and the electric readout system. We notice the negative sign of the\nflux locked loop (FLL) feedback, due to periodic transfer function (sin-like) of the\nSQUID, the FLL is stable whatever the sign. Indeed, If the feedback is positive the\nsystem is unstable and thus the operating point of the SQUID jump to the other\nslope of the SQUID characteristic (the transfer function HSQ become −HSQ). So\nthat the periodic SQUID transfer function leads to always operate in a negative\nfeedback which is in steady state. In addition, it is possible to invert the connection\nto the loops of the transformer coils: Min → −Min and/or Mfb → −Mfb. So that,\nin the end, we do not really know exactly the sign, but we are sure that the system\ncan find a stable state.\n\nhttps://github.com/satorchi/pystudio\n\n\nThe cryostat and the electronic readout system 168\n\nG LNA\n\n[V ]\nH sq+\n\n[ A]−V\nR2\n\nV\n\nα\nR0\nT 0\n\nP i\n\nP j\n\n[T ] [R]\n\n[ A]\n\n1\nG\n\nM i n +\n[V ]\n\n1\nR fb\n\n[ A]\nM fb\n\nV out\n\n[ ϕ0]ϕ fb\n\n_\n[ ϕ0] [ ϕ0]\n\n[ ϕtot ]\n\nFigure 5.20: The block diagram of a TES and the electric readout system\nwhich included the SQUID, the low noise amplifier, the PID controller and the\n\nflux locked loop feedback.\n\nFigure 5.21 shows the output signal of I-V measurements for a 256 TESs array.\nThere are 2 ASICs in the interface (green-ASIC2 and blue-ASIC1 color online),\nwe can observe number good or bad pixels.\n\nA DC-SQUID consists of two Josephson junctions5 connecting parallel on a su-\nperconducting loop. It is extraordinarily sensitive to convert magnetic flux to\nvoltage(V-Φ) and current voltage. A fundamental property of a closed super-\nconducting loop is that they can maintain the magnetic flux in a universal con-\nstant called the flux quantum. A flux quantum can be understood as the ra-\ntio of Planck constant h and the absolute value of the Cooper-pair charge 2e,\nΦ0 =\n\nh\n\n2e\n≈ 2× 10−15 Wb [1, 122].\n\nThe current passes by a SQUID, normally it is divided equally by two 2 × I1 =\n\n2 × I2 = Isq = I0 sin ∆φ, where ∆φ is the different phase. The output voltage of\na SQUID is a periodic function of magnetic flux, this is explained in the super-\nconducting electromagnetic theory of the relationship between phase change and\nwavefunction in canonical momentum. Hence we could measure the magnetic sig-\nnal by an integer number of flux quanta sensitively. If we applied a magnetic field\nto a SQUID, the superconducting loop opposes the magnetic field by generating a\nscreening current Is, which is equal but opposite the applied magnetic field and the\n5Josephson junction: weak link thin insulator or normal regions of superconductors, the current\ncan flow with zero resistance called Josephson current, the maximum current before having\nresistance is called the critical current (Ic).\n\n\n\nThe cryostat and the electronic readout system 169\n\nFigure 5.21: The read out result of the QUBIC studio tool and read out\nsystem for a 256 TESs array, half-bottom pixels (green) are read by ASIC2 and\nthe half-top pixels(blue) are read by ASIC1. The curves in the pixels are the IV\n\nmeasurements for each TES.\n\n\n\nThe cryostat and the electronic readout system 170\n\nscreening current cancels the flux in the Josephson junctions ring. This current\nis periodic in the applied flux, the critical currents of Josephson junctions depend\non the screening current [1].\n\nI1 =\nIsq\n\n2\n− Is,\n\nI2 =\nIsq\n\n2\n+ Is. (5.39)\n\nWhen we applied a magnetic flux in the order of one flux quantum to one and a\nhalf flux quantum, this is equivalent to increase continuously the magnetic flux\nfrom zero to a determined value. The SQUID I-V curve oscillates between two\nregimes with a period of one flux quantum. This physical concept helps to make a\nmagnetometer detector. The input flux and the output voltage across the SQUID\nlook sine-like oscillation as shown in figure 5.22.\n\nIn fact we use SQUIDs with a feedback coil in a Flux-Locked Loop (FLL) [122],\nit means the current from SQUID will be fed again the second coil of the SQUID.\nConsequently the negative feedback can compensate external flux to help operating\nsystem in the constant flux between Φin, Φfb.\n\nThe FLL technology can measure tiny voltage-bias using a SQUID. The principle\nis that when we apply an unknown voltage to one of SQUID coils, the screening\ncurrent appears, the FLL coil will create a canceling magnetic flux of the unknown\nvoltage. The current of the FLL is measure then the voltage is determined by a\nresistor Rfb. The block diagram in figure 5.20 and 5.23 illustrate the FLL opera-\ntion. The relationship among the voltage, the feedback resistor, the transformer\ncoil, and original from a TES is:\n\nVout = G×KPID × VSQUID,\n\nΦin = MinIin,\n\nΦfb = MfbIfb =\nMfb\n\nRfb\n\n× Vout. (5.40)\n\nIf the FLL loop gain is � 1 then:\n\nΦin − Φfb = 0,\n\nMinIin =\nMfb\n\nRfb\n\n× Vout. (5.41)\n\n\n\nThe cryostat and the electronic readout system 171\n\nFigure 5.22: (top) A microphotograph of the SQUID pins scheme, glued and\nbonded on a printed circuit board (PCB) for a flux locked loop operating in su-\nperconducting temperature.(bottom) The sine-like SQUID characteristic output\nvoltage as function of magnetic flux Φin. The different lines are obtained by\n\ndifferent currents bias Isq [31].\n\nIn the language of an automatic controlled system, the block diagram of the TESs\nclose-loop feedback readout system is model as figures 5.23 and 5.20:\n\n• Hsq =\ndVsq\n\ndΦtot\n\nis the dynamic transfer function of the SQUID.\n\n• G is the gain of the cryogenic SiGe followed by a preamplifier Stanford\nResearch SR560 at room temperature providing a total gain G = 200000\n(GSiGe=200,GSR560 = 1000).\n\n\n\nThe cryostat and the electronic readout system 172\n\nbias\nG\n\nVout\n\nRfb\nφfb\n\nφext\n\n≡\nφext\n\nHsq G KPID\nVout\n\n1\nRfb\n\nMfb\nφfb\n\nφtot\n-+\n\n[φ0] [φ0] [V] [V]\n\n[A][φ0]\n\nFigure 5.23: SQUID readout system and its corresponding block diagram in\na Flux Locked Loop. This block diagram is associated with the block diagram\n\nin figure 5.20\n\n• KPID is the transfer function of the PID controller, which is embedded in the\nFPGA board. In practice, we used to configure PI or I controllers.\n\n• Rfb is the feedback resistor of the FLL, Rfb = 10 kΩ or 110 kΩ.\n\n• Mfb is the transformer coil of the FLL.\n\nThe electrical time constant of the readout chain\n\nWe can study the stability of the system in the Laplace domain6, the transfer\nfunction of the PID controller is expressed L (s) = KPID = KP +\n\nKI\n\ns\n+ sKD =\n\nKI + sKP + s2KD\n\ns\nwhere s is the complex frequency. The transfer function of the\n\nclose loop, negative feedback system is calculated in Laplace domain:\n\nFB =\nHsqGKPID\n\n1 + HsqGKPID\nMfb\n\nRfp\n\n,\n\n=\nHsqG\n\nKI + sKP + s2KD\n\ns\n\n1 + HsqG\nMfb\n\nRfp\n\nKI + sKP + s2KD\n\ns\n\n,\n\n=\n\nHsqG\n\n(\n1 +\n\nKP\n\nKI\n\ns +\nKD\n\nKI\n\ns2\n\n)\n\ns\n\nKI\n\n+ HsqG\nMfb\n\nRfb\n\n(\n1 +\n\nKP\n\nKI\n\ns +\nKD\n\nKI\n\ns2\n\n) ,\n\n=\nRfb\n\nMfb\n\n1 +\nKP\n\nKI\n\ns +\nKD\n\nKI\n\ns2\n\n1 +\nKP\n\nKI\n\ns +\nRfb\n\nHsqGMfbKI\n\ns +\nKD\n\nKI\n\ns2\n\n. (5.42)\n\n6it transfers a variable from real domain to the complex domain. Normally it is transformation\nof time to frequency.\n\n\n\nIV, VP, VR curves 173\n\nDue to the fact that we do not often design a full three PID components system.\nWe use to set the derivative KD = 0. Then we can define responded time constants\nof the system in the equation 5.42\n\n\n\n\n\nτ0 =\nRfb\n\nHsqGMfbKI\n\n+ τ01,\n\nτ01 =\nKP\n\nKI\n\n.\n\n(5.43)\n\nEspecially, the QUBIC studio has separated scripts, which are compensated signal\nto the zero baseline. Consequently we also set the proportional KP = 0. Finally\nthe electronic readout time constant of the system is depended only on the integral\nKI term of the PID controller since we know Rfb, Hsq, G,Mfb [122]:\n\nτ0 =\nRfb\n\nHsqGMfbKI\n\n. (5.44)\n\n5.3.1 IV, PV, RV curves\n\nThe TES Current-Voltage characteristic (I-V curves) represents the TES operating\ncurrent, voltage, resistance..., as a function of bias voltage in the electrothermal\nfeedback (ETF) regime. The I-V curves measurements also establish a relationship\nbetween the voltage and the current of a TES, particularly in the TES transition\nregime. The measurements of TES I-V curves have a vital role for several reasons\n[156]:\n\n• The I-V curves help us to determine saturation energy of the pixels, the\nmaximum dissipated energy of the TES without driving the TES to the\nnormal regime.\n\n• The I-V curves provide information such as the TES operating current and\nresistance as a function of bias voltage, when I-V curves are taken at multiple\nbath temperatures, the thermal conductance of the TES to the thermal bath\nhelps determine the TES behaviors such as the TES noise and responsivity\nand the optimal TES operating condition.\n\n\n\nIV, VP, VR curves 174\n\n4 5 6 7 8\nVTES [ V]\n\n5\n\n6\n\n7\n\n8\n\n9\n\n10\n\n11\n\nI T\nES\n\n[\nA]\n\n2\n6\n7\n15\n18\n23\n25\n26\n28\n42\n49\n50\n63\n64\n\n4 5 6 7 8\nVTES [ V]\n\n20\n\n30\n\n40\n\n50\n\n60\n\nP T\nES\n\n[p\nW\n\n]\n\n2\n6\n7\n15\n18\n23\n25\n26\n28\n42\n49\n50\n63\n64\n\n4 5 6 7 8\nVTES [ V]\n\n0.4\n\n0.5\n\n0.6\n\n0.7\n\n0.8\n\n0.9\n\n1.0\n\n1.1\nR T\n\nES\n[\n\n]\n2\n6\n7\n15\n18\n23\n25\n26\n28\n42\n49\n50\n63\n64\n\nFigure 5.24: I-V, V-R and P-V curves of the several pixels on a 256-TES at\nTbath = 300 mK. We apply changing voltage-biased source and measure the\n\noutput current.\n\n• The I-V curves also help to calibrate the TES complex impedance and the\nTES noise spectrum, which are necessary for establishing an electrothermal\nmodel of the TES.\n\nFigure 5.24 represents I-V, R-V and P-V curves of several pixels on a 256 TES\narray. A typical TES I-V curve is divided into three states: the normal state, the\ntransition state, and the superconducting state. First I-V curves describe that\nfrom the right part of the curves, the TESs are in their normal state following the\nmetallic behavior of Ohm’s law. Then, the TESs tend to transit to their supercon-\nducting state, and the electrothermal feedback starts to take place when the I–V\ncurves reach their minimum. Once the electrothermal feedback is operational, the\nTESs are self-regulated and work at quasi-constant power, which corresponds to\nthe P–V plateau. As the TESs travel further on their transition, their resistance\ncontinues to drop, which leads to I–V portions of a parabola on the left part of\n\n\n\nIV, VP, VR curves 175\n\nthe curves. The flat region of the P-V curve shows that the total dissipated power\nis a constant regardless of changes in the voltage bias which is an evidence of an\nelectrothermal feedback regime [93]. The R-V curves represent for currents in the\nTESs, TESs operate in the normal regime obey Ohm’s law, they transit to the\nsuperconducting regime, a small change of the resistance equivalents to a large\nchange of the current then these curves are not linear.\n\n0 2000 4000 6000 8000 10000 12000\nt [sample]\n\n4.0\n\n4.5\n\n5.0\n\n5.5\n\n6.0\n\n6.5\n\n7.0\n\n7.5\n\n8.0\n\nV b\nia\n\ns[\nV]\n\n0 1000 2000 3000 4000 5000 6000 7000 8000\nt [sample]\n\n0.0\n\n0.5\n\n1.0\n\n1.5\n\n2.0\n\n2.5\n\nI T\nES\n\n[\nA]\n\nFigure 5.25: left: The input sine Vbias signal, which is sent to the TES. right:\nThe output signal of the readout system when the TES array is cooled down to\nthe superconducting temperature. The grey area indicates the TES signal in the\ntransition regime while outside the grey area, the TES signal is in the normal\n\nregime.\n\nIn addition, in order to estimate the resistance of the TES below Tc, practically\nwe vary the TES bias voltage following very low-frequency sine waves (0.1 Hz)\nwe change decreasingly the temperature of the cryostat to the superconducting\nregime. We can observe the behavior of the signal between the normal state and\nthe superconducting state by result of the changing output signal in QUBIC studio\ntool as shown in figure 5.25. The resistance values of the TES is calculated using\nthe formula 5.8:\n\nRTES =\nVbiasRpara\n\nRbias + Rpara\n\n1\n\nNpts∆SsigG\nMfb\n\nMinRfb\n\n. (5.45)\n\nThe left term of the equation\nVbiasRpara\n\nRbias + Rpara\n\nis the TES voltage as described in the\n\nprincipal circuit of a TES while the right term is\n1\n\nITES\n\nfrom the electronic readout\n\nsystem with ∆Ssig is the dynamic (amplitude with no offset) of the output which\nis measured in the sample unit. When a TES is in the normal state, the value\nof RTES is close to ≈ 1 Ω, figure 5.6 shows the curves of temperatures in K and\nresistances Ω and also the values of the sensitivity temperature parameter α for\n\n\n\nRadioactive source Americium 241 176\n\nsome TESs. When the TES enters the superconducting regime, the value of α is\nhigh and it decreases steadily in the normal regime. Figure 5.25 illustrates the\nchanging of the signal of a TES between the normal state and the superconducting\nstate. The voltage of the TES evolves faster in the superconducting state than in\nthe normal state while the Vbias follows a sine wave. This point has been figured\nout by analysis the SQUID signal as well as experiments. In the QUBIC studio\nsoftware, we can see the changed signal in the screen when the detectors enter the\nsuperconducting temperature.\n\n5.4 Radioactive source Americium 241\n\nThe 241Am7 half-life (t1/2 = 432.2 years) source has diameter 3 mm. It has an\nactivity of 8 Bq and the α particles are emitted with an energy of 5.44 MeV\n(λ ≈ 0.23 pm), the γ rays are emitted with an energy of 60 keV. The source is\nplaced at 5mm distance from a TES array which could be in front of a determined\npixel. In order to minimize the thermal background, we designed a copper holder\nwhich can be mounted to the bonded TES array. The copper holder is covered\nby copper tape which is placed in front of the TES array. On the copper cover,\nwe attach the radioactive source. The configuration of the experiment is shown in\nfigure 5.26.\n\nIn order to estimate simply the deposited energy of particles through pixel struc-\ntures, we can present an approximate formula:\n\n∆E = e\ndE\n\ndx\nρ [MeV], (5.46)\n\nwhere e and ρ are the thickness [cm] and the density [g/cm3] of materials, respec-\n\ntively. Firstly we calculated the stopping power\ndE\n\ndx\nof materials which made the\n\nTES pixel. We used SRIM8 with the range of incident energy from keV to GeV.\n7241Am radioactive source can be found in residential smoke detectors. 241Am emits α particles\nand γ rays. α particles or 4\n\n2He\n+ consists of two protons and two neutrons. γ rays are electro-\n\nmagnetic waves of very short wavelength. According to Public Health Statement of National\nTechnical Information Service (NTIS), Virginia USA, we may be exposed to 241 Am a little by\nbreathing air, drinking water or eating food. Alpha particles do not penetrate the skin and\ngamma rays emitted from americium sources are low energy. Then exposure to americium is\nnot usually considered to be a danger to human health. The 241Am source emitted α particles\nat ∼ 5.44 MeV, then the penetration is around 4 cm in air.\n\n8The Stopping and Range of Ions in Matter: calculates many features of the transport of ions\nin matter http://www.srim.org/\n\nhttp://www.srim.org/\n\n\nThe simple TES model approach 177\n\nFigure 5.26: Configuration of Americium source and the TES array for the\nexperiment. Top left : The americium source. Bottom right : The source is set\n\nup in font of the TES array in the cryostat.\n\nWith compound materials we assumed 50 % of each ingredient except the NbSi\nthat is taken 15.45 % of Nb. The result of stopping power is shown in figure 5.27.\n\nApply the equation 5.46 we obtain the deposited energy in the table 5.3. Due\nto the thickness of the substrate, the energy of alpha particles is absorbed in the\nsubstrate layer.\n\n5.5 TES model approach\n\nWe can model a basic TES pixel as a film of a heat capacity linked to a silicon\nwafer substrate and a cooling bath by thermal conductances. The temperature\nelevation of a detector is evaluated by the Joule heating and the losing heat to the\n\n\n\nThe simple TES model approach 178\n\n103 104 105 106 107 108 109\n\nE0 [eV]\n\n101\n\n102\n\n103\n\ndE dx\n[M\n\neV\ncm\n\n2 /g\n]\n\nAl\nIr\nNbSi\nNb\nSiN\nSi3N4\nSi\nTiV\n\nFigure 5.27: The stopping power of alpha particles in different materials, data\nis exported by SRIM software.\n\nElement Material Thickness [µm] Density [g/cm3] Deposit energy [keV]\nThermometer NbSi 0.1 6.8 33.495\n\nGrid TiV 0.1 5.0717 23.7374\nWires Al 0.2 2.7 30.7372\n\nMembrane Si3N4 0.5 3.44 109.538\nSubstrate Si 500 2.329 5400 (absorbed)\n\nTable 5.3: The table of the deposited energy of alpha particles in different\nmaterials and its thickness. Due to the thickness of Silicon substrate, the emitted\n\nenergy of α particles is absorbed.\n\nsubstrate. The Joule heating provides ETF to hold a constant temperature of the\nTES. An equilibrium state is set when the Joule heating matches the heat going to\nthe thermal bath. Figure 5.28 shows the heat capacities of TESs C2 and the silicon\nwafer substrate C1(J/K). G1 is the thermal link of the silicon wafer substrate to\nthe bath temperature Tb of the cryostat whereas Cb is considered infinite. Tb must\nbe below Tc. In order to solve the thermal model in a simple way, we assume\nG2(W/K) the effective thermal link of TESs with the silicon wafer, in this case,\nthe thermal conductance G2 is increased by the electrothermal feedback effect then\nthey are equivalent to the effective thermal conductance Geff ≈ G(1 + L ) in a\n\nTES. So that we have effective time constants τ1 =\nC1\n\nG1\n\nand τ2 =\nC2\n\nG2\n\n. Notice\nthat the definition is not the same as the natural time constant of a TES which is\n\n\n\nThe simple TES model approach 179\n\nτ =\nC\n\nG\n.\n\nFigure 5.28: The simple model of TESs array, TESs link to the silicon wafer\nsubstrate by a thermal link, and it is similar for the silicon wafer and the thermal\n\nbath.\n\nWe assume particles from the radioactive source with energy Ein and hitting the\npixel which could be absorbed by the grids, the substrate, the thermometer . . . .\nThe deposited energy will induce temperature variations to the thermometer NbSi\n(∆T =\n\nE\n\nC\n) and effect on the TES signal. We define ∆T1(t) and ∆T2(t) are the\n\nrising temperature on the silicon wafer substrate and TES, respectively. Following\nthe simple model in figure 5.28 and the thermal equation for the temperature 5.3,\nwe have the thermal saturation of the silicon wafer and TESs over time are:\n\n\n\n\n\ndE1\n\ndt\n= C1\n\nd∆T1\n\ndt\n= G2 (∆T2(t)−∆T1(t))−G1∆T1(t).\n\ndE2\n\ndt\n= C2\n\nd∆T2\n\ndt\n= G2 (∆T1(t)−∆T2(t)) .\n\n(5.47)\n\nwe perform in the companion matrix form of the thermal equations.\n\nd\n\ndt\n\n\n\n\n∆T1(t)\n\n∆T2(t)\n\n\n =\n\n\n\n\n−G1 +G2\n\nC1\n\nG2\n\nC1\n\nG2\n\nC2\n\n−G2\n\nC2\n\n\n\n\n\n\n\n∆T1(t)\n\n∆T2(t)\n\n\n . (5.48)\n\nThis matrix equation is similar form with the linear algebra equation y′ = Ay,\nthen the general solution of the equation is:\n\n\n\n\n∆T1(t)\n\n∆T2(t)\n\n\n = c1 (k1) eλ1t + c2 (k2) eλ2t\n\n\n\nThe simple TES model approach 180\n\n= c1\n\n\n\n\nG2\n\nC1\n\nλ1 +\nG1 +G2\n\nC1\n\n\n\neλ1t + c2\n\n\n\n\nG2\n\nC1\n\nλ2 +\nG1 +G2\n\nC1\n\n\n\neλ2t.\n\nor = c1\n\n\n\n\nλ1 +\nG2\n\nC2\n\nG2\n\nC2\n\n\n\n\neλ1t + c2\n\n\n\n\nλ1 +\nG2\n\nC2\n\nG2\n\nC2\n\n\n\n\neλ2t. (5.49)\n\nwhere c1,2 are constants, k1,2 are eigenvectors ((A− λI) k = 0), we can choose 2\nvectors which satisfy the solution. λ1,2 are eigenvalues (det (A− λI) = 0) which\nbasically inverse of time constants.\n\nλ1,2 = −\n\n(\nG1 + G2\n\nC1\n\n+\nG2\n\nC2\n\n)\n±\n√\n\nDelta\n\n2\n;\n\nDelta =\n\n(\nG1 + G2\n\nC1\n\n+\nG2\n\nC2\n\n)2\n\n− 4G1G2\n\nC1C2\n\n. (5.50)\n\nWe define Λ1,2 = λ1,2 +\nG1 +G2\n\nC1\n\nthen we can rearrange the solution equations\n5.49, notice that Λ1 − Λ2 = λ1 − λ2.\n\n\n\n\n∆T1(t)\n\n∆T2(t)\n\n\n =\n\n\n\n\nc1\nG2\n\nC1\n\neλ1t + c2\nG2\n\nC1\n\neλ2t\n\nc1Λ1eλ1t + c2Λ2eλ2t\n\n\n . (5.51)\n\nIn order to find those constants c1,2, we have to consider the initial condition,\nsimply and firstly, we can assume α particles hit the substrate silicon wafer at\ntime t = 0, ∆T1 (t) =\n\nEin\n\nC1\n\n, ∆T2 (t) = 0. The equation 5.51 is now\n\n\n\n\nEin\n\nC1\n\n0\n\n\n =\n\n\n\n\nc1\nG2\n\nC1\n\n+ c2\nG2\n\nC1\n\nc1Λ1 + c2Λ2\n\n\n . (5.52)\n\n\n\nGlitch data analysis 181\n\nWe get \n\n\n\nc1 = −Ein\n\nG2\n\nΛ2\n\nΛ1 − Λ2\n\n,\n\nc2 =\nEin\n\nG2\n\nΛ1\n\nΛ1 − Λ2\n\n.\n\n(5.53)\n\nSubstituting to the equation 5.51 we have\n\n\n\n\n∆T1(t)\n\n∆T2(t)\n\n\n =\n\n\n\n\nEin\n\nC1\n\n1\n\nλ2 − λ1\n\n(\nΛ2eλ1t − Λ1eλ2t\n\n)\n\nEin\n\nG2\n\nΛ1Λ2\n\nλ2 − λ1\n\n(\neλ1t − eλ2t\n\n)\n\n\n\n. (5.54)\n\nSecondly we also can assume α particles hit the TES at the time t = 0, ∆T2 (t) =\nEin\n\nC2\n\n, ∆T1 = 0. Then we obtain\n\nc1 = −c2 =\nEin\n\nC2\n\n1\n\nΛ1 − Λ2\n\n. (5.55)\n\nsubstituting to the equation 5.51 we have\n\n\n\n\n∆T1(t)\n\n∆T2(t)\n\n\n =\n\n\n\n\nEin\n\nC2\n\nG2\n\nC1\n\n1\n\nλ1 − λ2\n\n(\neλ1t − eλ2t\n\n)\n\nEin\n\nC2\n\n1\n\nλ1 − λ2\n\n(\nΛ1eλ1t − Λ2eλ2t\n\n)\n\n\n\n. (5.56)\n\nThe equations 5.54 and 5.56 represent for the time constants of the interaction of\nparticles with a pixel including the silicon wafer substrate and the TES thermome-\nter. Notice that we are interested in the TES rising temperature ∆T2 (t) while ∆T1\n\nis important for the thermal cross-talk. In order to estimate the time constants of\nthe QUBIC’s TES array using those equations above, the table 5.4 presents the\nheat capacity and the thermal conductance of a TES pixel which considers as the\nsummation of the thermometer, the absorbing grid, and the membrane. We also\ninvestigate the ratio\n\nC\n\nG\n.\n\n\n\nGlitch data analysis 182\n\nElement Material C [pJ/K] G [pW/K]\nC\n\nG\n[ms]\n\nA pixel NbSi + TiV + Si3N4 3.79 140 ± 70 27 (18-54)\n\nTable 5.4: The table of materials, a pixel has three main components: The\nthermometer (NbSi), the metallic absorber grid and the membrane Si3N4 [94].\n\n5.6 Glitches data analysis\n\nIn this section, I describe the data analysis of glitches induced by interaction\nof α particles. In order to fitted glitches, I introduce an exponential function\ntemplate of an amplitude a and two time constants: the electric readout system\ntime constant τ0 and the thermal time constant τ1. Due to the fact that, the\nintrinsic electrical time constant of TES τel =\n\nL\n\nRL + RTES\n\n∝ 600 nH\n\n20 mΩ + 1 Ω\n≈ 0.6µs\n\nis too small then it is neglected. I propose an interpretation of time constants. In\naddition, I present the dependence of (i) the electric readout system time constant\nτ0 and the Flux Locked Loop, (ii) the voltage bias Vbias and time constants. Finally\nthe thermal cross-talk among TES pixels and the electronic readout chain cross-\ntalk are studied.\n\nFirst, we consider that when a particle hits the pixel, it can be either (1) the\nsilicon wafer substrate, (2) the absorbed grid or (3) the NbSi thermometer. In 3\ncases, the deposition of energy causes on increase of temperature of the medium\nwhich at the end induces a signal in the thermometer (TES). Figure 5.29 describes\nthat (a) the TES is in the transition region, the current follows in the RTES + the\ninductor L (the TES readout circuit is already described in figure 5.5), (b) when\nparticles hit the pixel, its energy is converted into heat transferred to the TES,\n(c) the small change in the temperature leading to the changing resistance of the\nthermometer, due to the value of Rshunt is as small as the value of Rpara then\nthe current follows to the Rshunt during very short time. This changing signal\ncontributed to the signal of the SQUID as well as the readout system. When the\nelectrothermal feedback and thermal bath set again the equilibrium state, (d) the\nTES returns to the superconducting transition state.\n\nFirst, we performed some measurements with the array P63, the frequency acqui-\nsition was set at 156.25 Hz, equivalent to a sample rate of 6.4 ms per bin sample on\nthe time order data (TOD). The data was mainly taken during around 4 minutes\nat different bath temperatures, thus we can study the I-V curves measurements.\nWe also vary the PID controller parameters of the electric readout system. After\n\n\n\nGlitch data analysis 183\n\nFigure 5.29: A single particle (Ex: an alpha particle, a single photon, cosmic\nrays . . . ) hits the pixel and behaviors of the electric system to the output signal.\n\nCredit: Jennifer Anne Burney [23].\n\nthe P63, we performed other measurements with the array P73, the frequency\nacquisition was set at 1562.5 Hz (0.64 ms). The data was collected during 10\nminutes, several sets of data are collected for an hour. Due to the fact that the\nfrequency acquisition is high then we have a huge amount of data to store. We\nhave chosen to work on 10 minutes of data to ensure a stability of the behavior\nand which are compatible with data analysis on a personal computer. Table 5.5\ngives data information of 27 times collected data. We are going to present the\nobtained results with the array P73 as the results. The array P63 has obtained\nsimilar conclusions.\n\nThe cryostat system is cooled down to 350 mK, the value of Rfb is 110 kΩ, applying\na voltage-biased. The TESs array is in the superconducting transition regime.\nWhen the readout system is well calibrated, glitches started to be observed. In\norder to understand the behavior of TESs, we will measure its time constants\ndepending on the electronic readout system parameters, especially the integral\nterm of the PID controller, the KI parameter is varied with a fixed voltage-bias.\n\n\n\nGlitch data analysis 184\n\nArray Frequency acquisition Number Remark\nof runs\n\nTOD ∼ 4’, 30’\nP63 156.25 Hz (∼ 6.4 ms) 27 Different bath temperatures\n\nDifferent PID controller parameters\nTOD ∼ 10’, 1h\n\nP73 1562.5 Hz (∼ 0.64 ms) 27 Different PID controller parameters\nDifferent voltage-biased parameters\n\nTable 5.5: The table of measurements data.\n\nIn contrast, the voltage-bias (Vbias) is varied with a fixed KI parameter. After\nthe voltage bias, there is a bias resistance Rbias = 10 kΩ, then the bias current is\napplied to a TES as shown in the practical TES bias circuit figure 5.5. Finally\nthe voltage applied to the TES is VTES =\n\nRL\n\nRbias\n\nVbias which is of the order of\nµV . Due to the position of the radioactive source compared with the TES array,\nseveral pixels are investigated to analyze the data. The study of time constants\nnot only helps to interpret the interaction of particles with pixel components but\nalso determines the best PID controller parameter for the TESs electronic readout\nsystem of the QUBIC experiment.\n\nFigure 5.30, we show ≈ 10 minutes timeline of the data with glitches and noise\nfor 2 typical pixels close to the radioactive source. The pixel 88 is in front of\nthe radioactive source. The baseline is set to zero by applying the median of\nthe original data. The y-axis shows the current in nano Ampere while on the\nx-axis is the bin sample. The sample rate is calculated using formula 5.38. This\nis a conventional parameter, according to the setting readout system 128 pixels,\n2 MHz readout for 10 samples, it derives that a bin is equivalent to 0.64 ms.\nFigure 5.31 presents the data processing of glitches. From the original time order\ndata, we applied methods for glitches detection, and glitches processing. After\nthat, a template model is used to fit the glitches signal. The time constants of\nglitches provide interpretations of particles on pixel components. The study of the\ncrosstalk among pixels is also addressed.\n\n5.6.1 Glitches detection\n\nIn order to detect glitches (the fast signal compare to CMB), the noise is assumed\nto have a Gaussian distribution. We have been carried out three methods. In\nthe first method, when we detect events greater than the threshold of the mean\n\n\n\nGlitch data analysis 185\n\nFigure 5.30: The glitches data of 2 TES pixels the 88 (in front of the ra-\ndioactive source) and the 69 in the sample unit (time 0.64 ms) and the current\n\namplitude I in nano-Ampere.\n\nand 3 times the standard deviation µs + 3σs of the data, we consider that events\nare glitches. However, this first method has the disadvantage of small glitches\ndetection even though we have a fine-tuned threshold. The second method is an\nimprovement of the first method, we use the same threshold of µs +3σs of the data.\nAfter that we separate the noise signal and the glitches signal, we only calculate\nthe mean µn and the standard deviation σn of the noise signal, iteratively we\napplied the threshold of µn + 3σn for the whole data again to detect the glitches.\nNevertheless, the fluctuations of the data signal in the small scales lead to wrong\ndetected glitches. Finally, the third method is chosen to present the results, the\ndata processing follows as:\n\n\n\nGlitch data analysis 186\n\nGlitches DetectionGlitches Detectionμ ,σ\n\nGlitches\nProcessing\n\nCrosstalkCrosstalk\n\nTODTOD\n\nFit GlitchesFit Glitches\nχ2 ,σn\n\nGlitches\nBaseline\nGlitches\nBaseline\n\nInterpretation\nTime constants\nInterpretation\n\nTime constants\n\nMaximum\nCorrection\nMaximum\nCorrection\n\nStacking\nMethod\n\nStacking\nMethod\n\nMedian\nMethod\nMedian\nMethod\n\nFigure 5.31: The flow chart of glitches data analysis. TOD stands for the\ntime order data. µ, σ are the mean and the standard deviation of the data. χ2\n\nmethod is used to estimate the best fit associated with the standard deviation\nof the noise σn.\n\n\n\nGlitch data analysis 187\n\n• The baseline of the whole data is processed using a median window of 2000\nbin samples. This method is so-called the median filter.\n\n• The mean and the standard deviation of the signal are calculated µs , σs.\n\n• The stated threshold to detect glitches is the mean and 2 times the standard\ndeviation µs + 2σs. This threshold means that we will detect 5% which are\nnot glitches. The next step solves this issue.\n\n• When a glitch event is detected at a level greater than µs + 2σs, we compare\nthe value of event with the local noise of a 100 bins sample window, iteratively\nthe threshold of the mean noise µn pluses 3σn of the local noise is applied to\ndetermine glitches. This threshold implies that 99.7 % of noise is rejected,\nit means that 3 events will be detected as glitches due to the noise or signal\nfluctuations in a total of 1000 detected events.\n\n• A sliding window of 750 bin samples is taken to study the glitch, starting\n200 bin samples before the position of maximum glitch. This window is\nequivalent to 0.48 second. This choosing window takes into account the\nnumber of glitches over the timeline. In general, we have about 200 detected\nglitches for 10 minutes, it corresponds to 1 glitch per 3 seconds.\n\n• The amplitude of a glitch is chosen as the maximum peak value of the sliding\nwindow data. The peaks of detected glitches are stored as well as their bin\npositions. Then the rising time of a glitch is defined by 200 bin samples and\nthe decay time of a glitch is defined by 550 bin samples. The local noise is\nalso taken for a window of 100 bins sample starting from the beginning of\nthe window.\n\nFigure 5.32 shows the distribution of the data with the threshold 2σs and 3σs for\ntwo typical pixels. The position of the maximum of the distribution is close to\nzero because of the baseline signal resulting from the median filter method. The\ntails is due to the glitches signal.\n\nBecause the signal has small scales fluctuations in the timeline, the noise can vary\nover the threshold. A similar detection method is applied for the inverted data\nto evaluate the level of noise. After detecting peaks, the mean of the local noise\nis calculated. The absolute amplitude of a glitch is computed by subtracting the\npeak and the mean of the local noise. Figure 5.33 shows the histograms of the\n\n\n\nGlitch data analysis 188\n\n10 0 10 20 30 40\nI [nA]\n\n10 1\n\n100\n\n101\n\n102\n\n103\n\n104\n\n105\n\nOc\ncu\n\nrre\nnc\n\ne\n\n = 0.276\n\n = 2.322\n\nrun7pix69\n\n20 10 0 10 20 30\nI [nA]\n\n10 1\n\n100\n\n101\n\n102\n\n103\n\n104\n\n105\n\nOc\ncu\n\nrre\nnc\n\ne\n\n = 0.268\n\n = 2.148\n\nrun7pix88\n\nFigure 5.32: The histogram of data for run7 detector 69 and detector 88.\nThe black vertical line is the mean µs value of the signal, the vertical blue lines\nindicate the 2σs of the signal, the vertical green lines indicate the 3σs of the\nsignal. The tail of the signal is the glitches data. The negative populations are\n\nthe noise signal.\n\nabsolute amplitude of detected glitches and the level of the noise contribution.\nThese peaks values indicate that the amplitudes of glitches peaks are in a range of\n∼ 10− 37 nA. There are two populations in the distribution. The high amplitude\nregion could be the contribution of α particles (Eα ∼ 5.4 MeV) interaction with\npixels while there is lower amplitudes beside the noise level that is a possible\ncontribution of gamma rays (Eγ ∼ 60 keV).\n\n0 10 20 30 40\nAmplitude [nA]\n\n0\n\n2\n\n4\n\n6\n\n8\n\n10\n\n12\n\n14\n\n16\n\n18\n\nOc\ncu\n\nrre\nnc\n\ne\n\nrun7pix69\nnoise\n\n0 5 10 15 20 25 30 35\nAmplitude [nA]\n\n0\n\n2\n\n4\n\n6\n\n8\n\n10\n\n12\n\n14\n\n16\n\n18\n\nOc\ncu\n\nrre\nnc\n\ne\n\nrun7pix88\nnoise\n\nFigure 5.33: The histogram of glitches peaks of the signal (the black color)\nand the inverted data which is represented by noise level (the pink color).\n\nGlitches baseline\n\nIn order to obtain a better evaluation of the glitches baseline, the data processing\nof detected glitches follows steps:\n\n\n\nGlitch data analysis 189\n\n1. To the signal of each detected glitch, we subtracted each bin value to the\nmean value of the detected local noise → Absolute glitches signal.\n\n2. After that, we subtracted again each bin value to the median of a window\nof 100 bins sample starting from the beginning of the window → Glitches\nbaseline.\n\n3. According to the peaks of the inverted data as shown in figure 5.33, a filtered\nnoise which is the median or the mean plus 4 or 5 σ of the negative peaks\ndistribution, is applied to reject the noise contribution in the signal. Only\nglitches with the amplitude of peaks greater than this filter threshold are\nconsider as glitches.\n\n4. Because small-scale fluctuations of the signal could happen around the maxi-\nmum of the glitches. We use a median window of 9 bins sample to determine\nagain the glitches maximum positions then we reprocess the glitches timeline\nwith corrected maximum positions.\n\nFigure 5.34 is an example of a typical glitch (left) and all glitches (right) after\nselection. The first impression is that glitches look to have similar shapes and\njust varying amplitudes, probably the glitches occur due to the interaction of the\nα particles, and the evidence of glitches due to γ particles are too difficult to\nrecognize because of the incident energy is very small and the noise is complex.\nThe baseline is not exactly stayed at the zero due to small shapes fluctuations\nof the noise signal (not flat), these fluctuations will be rejected when we average\nglitches to study time constants.\n\n0 200 400\nTime [ms]\n\n0\n\n10\n\n20\n\nI [\nnA\n\n]\n\n0 200 400 600\nTime [sample]\n\n0\n\n10\n\n20\n\n30\n\nI [\nnA\n\n]\n\nrun7pix88, 130 glitches\n\nFigure 5.34: An example glitch and all the 130 glitches after data processing\nfor the pixel 88 which is in front of the radioactive source.\n\n\n\nGlitch data analysis 190\n\n5.6.2 Fit glitches\n\nI will describe results obtained on the \"run7\" taken with voltage-bias of the TES\nVTES = 5.0 µV, and the PID controller KI parameter equals 1000. In order to\ninvestigate time constants of a glitch, we based on the simple model and the\nsolution which has been represented in the equations 5.54 and 5.56. A general\ntemplate model (equation 5.57) is used to fit the data. On the right-hand side of\nthe equation, the first term is the rising time constant of the glitch, the electrical\nreadout system time constant τ0, and the second term is the decay time constant\nof a glitch, the TES responding time constant τi of an event (the thermal time\nconstant).\n\nS(t) = a\n(\n1− exp−(t−t0)/τ0\n\n) n∑\n\ni=1\n\nexp−(t−t0)/τi +c. (5.57)\n\nHere a is the amplitude parameter of the glitch, t is the bin sample data corre-\nsponding with S(t) value. For the initial starting fit values, t0 is set at 100 bin\nsample (64 ms), the offset c is set to zero, the bound of amplitude a is a positive\nvalue.\n\nFigure 5.35 presents some fitted glitches, all fitted glitches are presented in the\nAppendix C. The χ2 estimator is calculate for each glitch. We consider the noise\nfor the χ2 estimation as the standard deviation of a window of 100 bins samples\nstarting from the beginning of the glitch window. The reduced χ2 will be used\nto determine the quality of the fit. We express the first exponential function of\nequation 5.57 in Taylor series, a\n\n(\n1− exp−(t−t0)/τ0\n\n)\n≈ a\n\nτ0\n\n(t− t0), and we can see\nthat we have the degeneration of the amplitude a and the electronic time constant\nof the readout chain τ0. The degeneration can be observed in parameters of the\nfitted glitches.\n\nFigure 5.36 shows the plot of estimated time constants versus glitches maximum.\nBecause of the degeneration of the a and the τ0, the maximum value of glitches\nare a good alternative way to study behaviors of glitches. The histogram of the\nmaximum has a mean of the order of 20 nA. The histogram of the decay time\nconstant τ1 has a mean of the order of 40 ms. The histogram distribution of the\nrising time constant τ0 shows two populations which help us to understand the\ninteraction of particles with TES pixel components.\n\nTo improve the global understanding of the glitches, we applied two methods to\ndetermine average glitches:\n\n\n\nGlitch data analysis 191\n\nFigure 5.35: Example of fitted glitches for pixel 88. The glitch identification\nis label g. Parameters of the fitted template model are labeled. The reduced χ2\n\nis used to evaluate the fit.\n\n• Stacking glitches method: For this mean method, all the glitches data\nare summed to become a final mean glitch when divided by the number of\nglitches.\n\n• Median glitches method: Each glitch is divided by the maximum value to\nobtain the normalized glitch. After that, the median of all glitches for each\nindividual bin data point is applied to obtain the final normalized glitch.\n\nFigure 5.37 presents the result of the stacking glitches method and the median\nmethod for all glitches in the run7 pixel 88 while figure 5.38 and 5.39 present for\nthe different populations of the electrical time constant of the readout chain τ0.\nThe stacking glitches method provides the estimation of the amplitude of the glitch\nwhile the median glitches method estimates more accurately the time constants\nvalues. In fact, the process of stacking small and big glitches together affects the\nrising edge of the glitch. Consequently, the baseline of the median glitches method\nis better than the baseline of the stacking glitches method. This can be proofed\n\n\n\nGlitch data analysis 192\n\n0\n\n10\n\n20\n\n30\n\n40\n\n50\n0\n1\n\n0\n\n10\n\n20\n\n30\n\n40\n\n50\nMaximum\n\n0 20 40 60 80 100\n\n0 20 40 60 80 100\n\nrun7pix88,130glitches\n\n0, 1 [ms]\n\nM\nax\n\nim\num\n\n [n\nA]\n\nFigure 5.36: Time constant and peak values of glitches. Two populations of\nthe rising electrical time constant τ0 are observed on the histogram.\n\nby the estimation of the χ2 value. Therefore, we will present time constants using\nthe median glitches method and the amplitude using the stacking glitches method.\n\nFigure 5.40 presents the results of the fit on the median glitches method for several\npixels. The table 5.6 shows the results for different pixels, the fitted time constants\nvalues τ0, τ1 are estimated using the median glitches method, and the amplitude\nusing estimated by the stacking glitches method. We will not talk about the\nmismatch fitted model values because sometimes the fitted model does not fit the\ndata mathematically. (i) On the case of the peaks of the glitches could be resulted\nof scattering operating points in the superconducting transition regime for different\npixels. When an event of high energy (big glitch) happens, the operating point\n\nshould transit to the normal regime (τ1 ∼\nC\n\nG\n), when the thermal bath and the\n\nstrong ETF operates the operating point is pulled back to the superconducting\ntransition regime (τ1 ∼\n\nC\n\nGL\n). Finally the thermal time constants decays with\n\nthe contribution of two regimes. This case represents a better fit on the peaks\nglitches. Furthermore the logarithmic sensitivity to temperature parameter α is\nsmall in the normal state, the loop gain parameter L is proportional to α. The\n\n\n\nGlitch data analysis 193\n\n0 100 200 300 400 500\nTime [ms]\n\n0\n\n5\n\n10\n\n15\n\n20\n\nI [\nnA\n\n]\n\nrun7pix88\n0=22.19±1.15 ms\n1=38.96±0.69 ms\n\na=48.02±1.85 nA\nt0=108.0±0.1 ms\nc=0.32±0.02 nA\n\n2=49.0; 2=0.065\nKI=1000\n\n0 100 200 300 400 500\nTime [ms]\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nNo\nrm\n\nal\niza\n\ntio\nn\n\nrun7pix88\n0=17.63±0.55 ms\n1=40.00±0.46 ms\n\na=2.32±0.05 nA\nt0=109.4±0.1 ms\nc=0.01±0.00 nA\n\n2=68.5; 2=0.002\nKI=1000\n\nFigure 5.37: Left: The stacking glitches method and Right: The median\nglitches method apply for all 130 glitches. These fitted parameters of the tem-\nplate model are labeled on the plot, together with the estimation of χ2 associated\nwith the standard deviation of the noise σχ2 . The value of the PID controller\n\nparameter KI equals 1000 for the run7.\n\n0 100 200 300 400 500\nTime [ms]\n\n0\n\n5\n\n10\n\n15\n\nI [\nnA\n\n]\n\nrun7pix88\n0=14.94±0.54 ms\n1=41.97±0.56 ms\n\na=23.34±0.73 nA\nt0=110.6±0.1 ms\nc=0.31±0.02 nA\n\n2=17.8; 2=0.090\nKI=1000\n\n0 100 200 300 400 500\nTime [ms]\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nNo\nrm\n\nal\niza\n\ntio\nn\n\nrun7pix88\n0=13.37±0.40 ms\n1=42.28±0.48 ms\n\na=20.89±0.03 nA\nt0=111.3±0.1 ms\nc=0.02±0.00 nA\n\n2=24.4; 2=0.004\nKI=1000\n\nFigure 5.38: The first population of the electrical time constant τ0: The stack-\ning glitches method (left) and the median glitches method (right) applied for\n\nthe 75 glitches.\n\ntime constant is the inverted proportion to L . Therefore, this case has a bigger\nvalue of the time constant, it also means that the fit is well fit. (ii) On the other\ncase, the low operating point, the adding transition is still in the superconducting\ntransition regime (τ1 ∼\n\nC\n\nGL\n), this case represents not well fit around peaks of the\n\nglitches. In addition, the different values of time constants in different pixels can\nbe explained by the readout of SQUIDs, due to the fact that each TES is read out\nby an independent SQUID. The characterized SQUIDs are not uniform including\nnoise performance also. Then the final outputs signal are not uniform among\npixels. This feature also can be observed by taking I-V calibrated measurements\nas shown in figure 5.24. The relative positions of pixels compared to the radioactive\n\n\n\nGlitch data analysis 194\n\n0 100 200 300 400 500\nTime [ms]\n\n0\n\n5\n\n10\n\n15\n\n20\nI [\n\nnA\n]\n\nrun7pix88\n0=42.69±4.46 ms\n1=34.31±0.88 ms\n\na=66.70±8.75 nA\nt0=105.3±0.1 ms\nc=0.20±0.02 nA\n\n2=12.2; 2=0.129\nKI=1000\n\n0 100 200 300 400 500\nTime [ms]\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nNo\nrm\n\nal\niza\n\ntio\nn\n\nrun7pix88\n0=37.19±2.80 ms\n1=35.04±0.71 ms\n\na=58.10±0.26 nA\nt0=105.6±0.1 ms\nc=0.01±0.00 nA\n\n2=10.3; 2=0.006\nKI=1000\n\nFigure 5.39: The second population of the electrical time constant τ0: The\nstacking glitches method Left and the median glitches method Right applied for\n\nthe 33 glitches.\n\nsource are different, hence the number of detected glitches has to be different.\n\nRun Vbias KI Pixel Glitches τ0 τ1 a\n(µV ) (ms) (ms) (nA)\n\n7 5 1000 69 80 11.41±0.15 64.31±0.33 33.28±0.38\n7 5 1000 70 54 7.89±0.16 35.91±0.27 54.25±2.08\n7 5 1000 75 77 23.07±0.95 53.64±0.84 37.01±1.28\n7 5 1000 81 94 23.76±0.35 72.97±0.43 32.15±0.70\n7 5 1000 87 70 10.17±0.13 47.11±0.22 36.52±0.53\n7 5 1000 88 130 17.63±0.55 40.0±0.46 48.02±1.85\n7 5 1000 93 75 15.99±0.36 43.33±0.36 51.14±2.14\n7 5 1000 106 23 35.57±1.34 71.84±1.05 86.54±15.32\n7 5 1000 107 73 60.72±9.75 39.01±1.35 103.74±60.06\n\nTable 5.6: The table indicates the number of glitches for several good pixels\nwith around 10 minutes recorded data which is labeled \"run7\" and the time\nconstants from the median glitches method for different pixels. The amplitude\n\nis computed with the stacking glitches method.\n\nFollowing the template model equation 5.57, a fitted model with a third or a\nfourth time constant does not work. We also fitted the only decay glitches with\nseveral time constants model. In addition, we make a template model with the\nrising time constant inside the decay time constant term, however, it was not\nconclusive. Therefore, we choose to have only one time constant for τ0, the rising\ntime constant of the electronic readout system and one τ1 for the decay time\nconstant (the thermal time constant). The value of the electric readout time\nconstant is typically 7-30 ms and the TES time constant is typically 20-60 ms.\nIn addition, we simply estimated the energy of the alpha particles in the range\n\n\n\nGlitch data analysis 195\n\n0 100 200 300 400 500\nTime [ms]\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nNo\nrm\n\nal\niza\n\ntio\nn\n\nrun7pix70\n0=7.89±0.16 ms\n1=35.91±0.27 ms\n\na=1.81±0.02 nA\nt0=118.4±0.0 ms\nc=0.02±0.00 nA\n\n2=16.2; 2=0.004\nKI=1000\n\n0 100 200 300 400 500\nTime [ms]\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nNo\nrm\n\nal\niza\n\ntio\nn\n\nrun7pix75\n0=23.07±0.95 ms\n1=53.64±0.84 ms\n\na=2.21±0.06 nA\nt0=102.9±0.1 ms\nc=0.03±0.00 nA\n\n2=25.7; 2=0.005\nKI=1000\n\n0 100 200 300 400 500\nTime [ms]\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nNo\nrm\n\nal\niza\n\ntio\nn\n\nrun7pix81\n0=23.76±0.35 ms\n1=72.97±0.43 ms\n\na=2.16±0.02 nA\nt0=100.9±0.1 ms\nc=-0.00±0.00 nA\n\n2=13.7; 2=0.004\nKI=1000\n\n0 100 200 300 400 500\nTime [ms]\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nNo\nrm\n\nal\niza\n\ntio\nn\n\nrun7pix87\n0=10.17±0.13 ms\n1=47.11±0.22 ms\n\na=1.79±0.01 nA\nt0=115.5±0.0 ms\nc=0.01±0.00 nA\n\n2=11.8; 2=0.003\nKI=1000\n\nFigure 5.40: The fitted parameters of the template model for different pixels\nin the same taken data run7. The KI parameter of the PID controller for run7\n\nis 1000 and the median glitches method is applied.\n\n2-5 MeV, in agreement with theoretical predictions. Our results demonstrate the\ncapability of our readout electronics in performing tests of sensitivity towards CRs\n[131].\n\n5.6.3 Interpretation\n\nWe can calculate the surface of a pixel components as shown figures 5.11 and 5.12.\nA pixel (TES thermometer + absorber + membrane) has:\n\n1. A surface of the absorber grid ∼ 2800 µm2 and ∼ 1µm thickness.\n\n2. Each square grid has dimension of 50× 50 µm hole and 5 µm width frame.\n\n3. The TES (NbSi) has a surface of ∼ 293 µm2.\n\n4. Between two pixels, the Si substrate has a surface of 2800× 200 µm and its\nthickness is 500 µm.\n\n\n\nGlitch data analysis 196\n\nFigure 5.41: The micro-photography of several pixels, the thermometer is at\nthe center of a pixel, the absorber is the metal grid, and the Silicon substrates\n\nare among pixels.\n\n5. The Al wires have a geometry of ∼ 10 cm length, ∼ 200 nm thickness, ∼ 6\nµm width.\n\nFigure 5.41 shows a micro-photography of a TES array and its components. The\nsurface of a pixel compared to the Si substrate is very high, however, the thickness\nof the pixel is very small compared to the Si substrate. In table 5.3 we estimate\nthe deposited energy by an α particle of 5.44 MeV versus the thickness of pixel\ncomponents. In general, if the particles hit one of the pixel components, the\ndeposited energy on the pixel is of the order of keV while the deposited energy on\nthe substrate is of the order of MeV .\n\nAs already said, the rising time constant τ0 estimation in figure 5.36 indicates two\npopulations of the rising time τ0 which is the electrical readout time constant for\nthe short one and could be a thermal time constant for the long one. According\nto the cross-section, the thickness and the deposited energy, the design, the fabri-\ncation and the assembly of a TES array, we can provide a hypothesis to interpret\nthis result:\n\n• In an equilibrium state, the Joule dissipation due to the voltage bias VTES\n\non a TES evacuates to the thermal bath (reference/fixed temperature).\n\n• The fist population: Particles hit directly to the sensor (thermometer TES\nor the absorber), thermal effect propagates very quickly to the thermometer\n\n\n\nGlitch data analysis 197\n\nand the rising time constant τ0 is the electronic readout time constant. The\nthermal equilibrium process is rapidly established due to the deposited en-\nergy on the absorber which has a thickness of 1 µm (τ1). A discussion of the\nthermal behaviour of the absorber and the TES thermometer can be found\non [19]. These events correspond to the first population.\n\n• The second population: Particles could hit the Si substrate, the deposited\nenergy is huge due to the thickness of 500 µm. To reach the sensor, the heat\nhas to affect the temperature of the TES through the thermal link between\nthis sensor and the Si substrate. The rising time constant should be close\nto the thermal time constant. Because the thermal coupling is not perfect\nbetween the Si wafer and the back copper (which is a better fixed thermal\nbath), there have several reasons affecting the thermal decoupling of this\ntwo layers. First, the assembly of the array is that the edge of the array is\nwell pressed over the back copper in order to hold the array with the back\nshort. The discussion of the applied force and thermal contact conductance\nbetween material layers can be found in the paper [132]. However the center\nof the array is not uniformly pressed over this copper thermal bath, then\nthe heat flows could transfer slower than the edge. Consequently, these heat\nflows arise the increment of the background reference temperature in which\nis finally detected by the sensor through a rising time τ0 dominated by the\nthermal time constant (more or less equal to τ1 ).\n\n• A proposed solution to fix this substrate thermal decoupling is that we can\nadd a gold layer on the back side of the Si substrate in order to fix and\nuniform the Si bulk temperature which thus could played better the role of\nthermal bath.\n\n• Space application: In the aspect of Cosmic Rays and a satellite’s focal plane\nusing TES arrays, the Silicon substrate surface plays an important role to\nreduce the impact of CRs.\n\nFollowing our hypothesis, we explain the two populations seen in the τ0 distribu-\ntion. The long τ0\n\n9 glitches are observed on the presentation of neighbor pixels at\nthe same time order data. Figure 5.42 presents an example of a glitch which has\nthe long time constant τ0. We observe that there have fluctuations of the signal of\nneighbor pixels. The observation reinforces the interpretation of the hypothesis.\n9The long τ0 means τ0 ≥ 32 ms\n\n\n\nTime constants and the KI parameter of the PID controller 198\n\n0\n\n20\ng=0 0=39.2±5.8ms\n\n1=50.6±2.4ms\na=98.5±7.6nA\n2=4.7 ; =0.5\n\nt0=93.2±0.3 ms\nc=-0.3±0.1 nA\n\npixel69 pixel70 pixel74\n\n0\n\n20\npixel75 pixel76 pixel81 pixel82\n\n0\n\n20\npixel86 pixel87 pixel93 pixel106\n\n1600 1800 2000\n\n0\n\n20\npixel107\n\n1600 1800 2000\n\npixel127\n\n1600 1800 2000\n\npixel128\n\n1600 1800 2000\n\nrun7pix88\n\nTime [ms]\n\nI [\nnA\n\n]\n\nFigure 5.42: The top left glitch is the glitch of the run7 pixel 88. This is the\nevidence of crosstalk\n\nIt means that particles hit the substrate, the heat is dissipated to among pixels.\nThe discussion of the crosstalk estimator is described in the section 5.7 below.\n\n5.6.4 Time constants and the KI parameter of the PID con-\n\ntroller\n\nAs described in the section 5.3, the KI parameter of the PID controller affect the\nrising time of the glitches as well as the amplitude of glitches. This demonstration\nis described in the transfer function equation of the readout system 5.44. KI\n\nparameter relates to the loop gain of the Flux Lock Loop (FLL) which changes\nthe bandwidth of the readout chain. When we increase KI parameter, the time\nconstant corresponding to the readout bandwidth must decrease. The equation 5.44\nis equivalent to y = 1/x form.\n\nTable 5.7 and figure 5.43 show very well how the time constants depend on the\nelectronic readout system KI parameter of the PID controller. Note that the value\n\n\n\nTime constants and the KI parameter of the PID controller 199\n\n0 100 200 300 400 500\nTime [ms]\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nNo\nrm\n\nal\niza\n\ntio\nn\n\nrun13pix69\n0=82.41±0.89 ms\n1=147.08±0.66 ms\n\na=2.79±0.02 nA\nt0=52.4±0.1 ms\nc=-0.00±0.00 nA\n\n2=0.0; 2=0.075\nKI=100\n\n0 100 200 300 400 500\nTime [ms]\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nNo\nrm\n\nal\niza\n\ntio\nn\n\nrun8pix69\n0=60.53±0.76 ms\n1=98.91±0.47 ms\n\na=2.90±0.03 nA\nt0=74.6±0.1 ms\nc=0.00±0.00 nA\n\n2=9.5; 2=0.003\nKI=200\n\n0 100 200 300 400 500\nTime [ms]\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nNo\nrm\n\nal\niza\n\ntio\nn\n\nrun12pix69\n0=20.60±0.26 ms\n1=68.35±0.35 ms\n\na=1.99±0.01 nA\nt0=101.7±0.1 ms\nc=0.01±0.00 nA\n\n2=12.0; 2=0.003\nKI=600\n\n0 100 200 300 400 500\nTime [ms]\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nNo\nrm\n\nal\niza\n\ntio\nn\n\nrun7pix69\n0=11.41±0.15 ms\n1=64.31±0.33 ms\n\na=1.63±0.01 nA\nt0=110.1±0.0 ms\nc=0.01±0.00 nA\n\n2=13.5; 2=0.003\nKI=1000\n\nFigure 5.43: The fitted model of time constants for the pixel 69 with different\nKI parameters of the PID controller with fixed voltage-bias VTES = 5µV. The\nKI parameter value is between 100, 200, 600 and 1000 as shown detail in the\ntable 5.7. The slope of rising time increase with KI according to the expected\n\nevolution of the readout bandwidth with KI .\n\nof KI parameter is an input on the QUBIC studio software to be changed by hand.\nIf we increase the value of the parameter KI , the time constant will decrease. The\nanalysis is shown for the pixel 69, but other analyzed pixels give similar results. A\nsimple evaluation of the deposited energy is given by the product of the amplitude\nand the time constant. The values are compared with the table 5.3 and we can see\nthat the values are compatible. However, this estimation is not accurately correct\nbecause we have to take into account the proportion of VTES to amplitude because\nof electrothermal feedback effects to the TESs voltage.\n\nFigure 5.44 shows the behavior of time constants respect to the changing KI\n\nparameter. It is possible to fit the plot of time constants versus KI parameter\nwith a y = 1/x function model. Because of the complexity of the electronic\nreadout system of TES technology, in particularly the QUBIC’s TES array, the\ncalibration parameters for a working stable system play an important role. As an\n\n\n\nTime constants and the voltage bias VTES 200\n\nRun Vbias KI Pixel Glitches τ0 τ1 a\n(µV ) (ms) (ms) (nA)\n\n13 5.0 100 69 76 84.62±0.97 148.57±0.69 22.75±0.37\n8 5.0 200 69 63 60.53±0.76 98.91±0.47 36.80±0.89\n9 5.0 300 69 73 39.22±0.60 88.87±0.56 29.03±0.31\n10 5.0 400 69 62 30.61±0.36 76.94±0.36 39.19±0.52\n11 5.0 500 69 73 23.73±0.22 70.13±0.27 43.75±0.64\n12 5.0 600 69 71 20.60±0.26 68.35±0.35 46.78±1.02\n7 5.0 1000 69 80 11.41±0.15 64.31±0.33 33.28±0.38\n14 5.0 1500 69 70 10.81±0.30 40.25±0.42 38.50±0.79\n\nTable 5.7: This table is given the measured time constants versus KI pa-\nrameters of the PID controller. Time constants are estimated using the me-\ndian glitches method while the amplitude is estimated using the stacking mean\n\nmethod.\n\n500 1000 1500\nKI parameter\n\n50\n\n100\n\n150\n\nTi\nm\n\ne \nco\n\nns\nta\n\nnt\n [m\n\ns] 0\n1\n\nFigure 5.44: Time constants respect to theKI parameter of the PID controller.\nThe voltage bias is fixed at 5.0 µV . The error bar is 1σ.\n\noutput of the study, in order to calibrate the QUBIC’s TES array, the value of KI\n\n≥ 1000 is a good parameter value for the electronic readout system. This value of\nthe KI parameter is also satisfied requirements of time constants of the QUBIC\nexperiment. For a higher value of KI , the FLL become unstable and extra noises\nstart to appear.\n\n\n\nTime constants and the voltage bias VTES 201\n\n5.6.5 Time constants, amplitude and the voltage bias VTES\n\nAs discussed before, from the equation 5.21, we know that the current responsivity\nof the superconducting TES is proportional to the inverse of the voltage bias\nsI ∼ −\n\n1\n\nVTES\n\n. It means also that if we increase the voltage bias, the electrical time\n\nconstants (τ0) will decrease. In practice, a study has been executed by varying the\nvoltage bias parameter VTES with a fixed KI parameter. In order to have many\ndata points, overall, the pixel 69 has been chosen as it was a good pixel for many\nruns and it is very close to the pixel 88 (in front of the radioactive source).\n\n0 100 200 300 400 500\nTime [ms]\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nNo\nrm\n\nal\niza\n\ntio\nn\n\nrun1pix69\n0=59.64±1.26 ms\n1=88.62±0.67 ms\n\na=3.03±0.05 nA\nt0=80.3±0.1 ms\nc=0.01±0.00 nA\n\n2=78.4; 2=0.001\nVTES=3.8 V\n\n0 100 200 300 400 500\nTime [ms]\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nNo\nrm\n\nal\niza\n\ntio\nn\n\nrun3pix69\n0=55.46±1.14 ms\n1=91.21±0.70 ms\n\na=2.86±0.05 nA\nt0=80.2±0.1 ms\nc=0.00±0.00 nA\n\n2=8.0; 2=0.005\nVTES=4.0 V\n\n0 100 200 300 400 500\nTime [ms]\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nNo\nrm\n\nal\niza\n\ntio\nn\n\nrun4pix69\n0=53.88±1.32 ms\n1=102.74±1.01 ms\n\na=2.61±0.05 nA\nt0=76.4±0.1 ms\nc=0.01±0.00 nA\n\n2=11.0; 2=0.006\nVTES=4.2 V\n\nFigure 5.45: The fitted model of time constants for the different voltage-bias\nVTES. The fixed KI = 200 parameter of the PID controller equals 200.\n\nFigure 5.45 shows the data and fitted curves for several runs of the pixel 69. In\nthe table 5.8 we give results of the behavior of the time constants in respect of the\nchanging voltage biased at a fixed KI = 200 parameter of the PID controller. The\nvoltage-bias ranged between 3.8 µV and 5.8 µV . The range of the voltage-bias is\nchosen based on the behavior of TES from a normal state to the transition regime\n\n\n\nTime constants and the voltage bias VTES 202\n\nstate and the superconducting state as shown on the study of I-V curves figure\n5.24, when we increase the voltage-bias, the TES array went close to the normal\nstate. Hence the TES time constants also could represent different meaning.\n\nRun VTES KI Pixel Glitches τ0 τ1 a\n(µV ) (ms) (ms) (nA)\n\n1 3.8 200 69 81 59.64±1.26 88.62±0.67 81.28±4.57\n3 4.0 200 69 73 55.46±1.14 91.21±0.70 36.49±1.00\n4 4.2 200 69 50 53.88±1.32 102.74±1.01 32.35±0.67\n5 4.5 200 69 78 75.90±2.24 89.25±0.86 71.89±3.61\n6 4.8 200 69 68 54.69±1.19 96.32±0.81 41.80±1.17\n8 5.0 200 69 63 60.53±0.76 98.91±0.47 36.80±0.89\n15 5.3 200 69 62 48.27±0.85 93.49±0.65 25.32±0.70\n16 5.5 200 69 37 41.25±0.73 122.08±0.97 11.38±0.18\n17 5.8 200 69 60 54.38±0.90 104.25±0.69 14.04±0.32\n\nTable 5.8: The table shows the relationship of time constants, the amplitude\nand the applied voltage-bias VTES. Time constants are estimated by the median\nglitches method while the amplitude is estimated by the stacking mean method.\n\n3.5 4.0 4.5 5.0 5.5 6.0\nVTES [ V]\n\n40\n\n60\n\n80\n\n100\n\n120\n\nTi\nm\n\ne \nco\n\nns\nta\n\nnt\n [m\n\ns]\n\n0\n1\n\n3.5 4.0 4.5 5.0 5.5 6.0\nVTES [ V]\n\n20\n\n40\n\n60\n\n80\n\nAm\npl\n\nitu\nde\n\n [n\nA]\n\na\n\nFigure 5.46: Left: Time constants versus voltage biases VTES. Right: The\namplitude versus voltage biases VTES. The grey color area means that TESes\n\nenter to the normal state. The KI is fixed at 200. The error bar is 1σ.\n\nFigure 5.46 presents the behavior of time constants, and amplitudes with respect\nto the changing voltage-bias. In general, when we increase the VTES, the thermal\ntime constant (τ1) increases because TES enters to the normal state, the logarith-\nmic sensitivity to temperature parameter α is small, the amplitude behavior has a\nnegative slope. Unfortunately, the data point at VTES = 4.5µV is affected by the\nchanging state of the TES from the superconducting state to transition regime.\nas we can deduce from the TES behavior at VTES = 4.5µV in I-V curves figure\n5.24. On the other hand, the PID controller parameter KI equals 200 is not an\n\n\n\nCross-talk 203\n\noptimal parameter for the electronic readout system as shown in figure 5.44. The\nbehavior of the electronic readout system changes very quickly at that regime,\nthis could affect the results and the slope of the relationship. In conclusion, this\nstudy would need more investigated data points as well as more fine-tuned PID\ncontroller parameter due to the fact that voltage-bias changes the state of the TES\nfrom a strong electrothermal feedback to a superconducting transition regime to\na normal regime.\n\n5.7 Cross-talk\n\nWhen a TES array is hit by α particles, cross-talk can happen at the stage of\nthermal dissipation or electronic readout chain.\n\n5.7.1 Thermal cross-talk\n\nWhat we call thermal cross-talk is the fact that a deposition of energy in a given\npixel could also induce temperature elevation of neighbors pixels. For thermal\ncross-talk analysis we fix a reference pixel, located in front of the radioactive\nsource, assuming that this pixel is the one mostly hit by α particles and then we\nobserve the signal of the neighboring pixels. Firstly we analyzed I-V curves of full\n256 TES pixels to identify the best radioactive source position which has many\ngood pixels around as shown in figure 5.47. The position of the radioactive source\nwas chosen in front of the pixel 88. What we wanted to evaluate is if after a particle\ninteraction in the central pixel, the temperature increase will propagate gradually\nfrom the red circle to the green circle and then to the blue circle, expecting that\nthe level of the thermal cross-talk will probably be related to the distance of pixels\ncompared with the pixel 88.\n\nFirst of all, figure 5.48 presents the TOD of several pixels at the same time during\n3.84 s. We can observe independent glitches occuring in different pixels. The\nexpression of the thermal cross-talk estimator is given in equation 5.58. Let us\nconsider a reference pixel where a glitch event happens: the glitch event g has\nthe bin position i of the maximum and the data of the maximum value of the\nreference pixel is d0\n\ne;i (in our study the reference pixel is the pixel 88). At that\nmaximum position i of a glitch, the data of all around pixels j is sum, however, the\n\n\n\nCross-talk 204\n\nFigure 5.47: The scheme of a 256 TES array, the IV curves represent good\nor bad pixels in an ASIC. The position of the Americium source is located in\nfront of the pixel 88 (the red circle) and the relative pixels around. The thermal\ncross-talk estimator C1 of round 1 (the green circle - pixels: 69, 76, 75, 87, 34,\n35, 93, 81) and C2 of round 2 (the blue circle - pixels: 128, 127, 74, 86, 33, 46,\n47, 48, 41, 94, 82, 70) are considered due to the distance of pixels respect to\nthe radioactive source position. The thermal cross-talk estimator C of the black\nsquare - whole pixel of the green circle and the blue circle - is considered for all\n\npixels except the pixel 88.\n\n861000 862000 863000 864000 865000 866000 867000\nTime [sample]\n\n10\n\n0\n\n10\n\n20\n\n30\n\nI [\nnA\n\n]\n\nrun7pix69\nrun7pix70\n\nrun7pix76\nrun7pix81\n\nrun7pix87\nrun7pix88\n\nrun7pix75\nrun7pix107\n\nFigure 5.48: 3.84 s TOD for several pixels which are the closest ones to the\nradioactive source.\n\n\n\nCross-talk 205\n\nlimited position has to consider the baseline then the baseline position b is given\nfor two positions which explain the factor of 0.5 in equation 5.58, statistically we\ncan divide by a smaller factor to obtain a high efficiency based on the variation\nconcept. All the glitches occurring in pixel 88 during the 10 minutes TOD are\nused and normalized by the number of glitches and the number of pixels. Finally,\nthe thermal cross-talk estimator depends on the number of glitches, number of\npixels, and the baseline. To estimate the thermal cross-talk, we try to evaluate if\nthere is a correlated signal in pixels close to the central pixel where glitches are\ndetected:\n\nC(b, pixel area) =\n1\n\nNg\n\n1\n\nNj\n\nNg∑\n\ng=1\n\nNj∑\n\nj=1\n\ndj\ni(g) − 0.5dj\n\ni(g)−b − 0.5dj\ni(g)+b\n\nd0\ni(g)\n\n. (5.58)\n\nIn this formula:\n- g is the glitch number on the reference pixel (pixel 88).\n- i(g) is the sample number where occurs the maximum of the glitch in pixel 88.\n- d0\n\ni(g) is the TOD value for the maximum of glitches in the reference pixel (pixel\n88).\n- dji is the TOD value of sample i of pixel jth (the reference pixel having j = 0).\n- b is a constant chosen to estimate the baseline of a given pixel TOD around the\ncentral sample defined as the glitch maximum position. The estimated baseline\nwill be given by\n\n(\ndj\n\ni(g)−b + dj\ni(g)+b\n\n)\n/2.\n\n- Ng is the number of glitches events in the reference pixel 88.\n- Nj is the number of pixels included in the set of pixels selected to evaluate the\ncross-talk:\n\n• Nj = 8 for the green circle corresponding to the pixels near the pixel 88.\n\n• Nj = 12 for the blue circle corresponding to the pixels external to pixel 88.\n\n• Nj = 20 if we consider all the pixels included inside the back square of figure\n5.47.\n\nFirstly, we applied different baseline positions b for the estimation of the baseline\nas a first step, we decided to study the level of cross-talk estimator for different\nround of pixels as shown in figure 5.47. The table 5.9 shows the values of cross-talk\nestimator in percent for different baseline positions b for the round 1, round 2 and\nround of all pixels in the black square as illustrated in figure 5.47 respect to the\n\n\n\nCross-talk 206\n\nBaseline position pixel C1(b)% C2(b)% C(b)%\nb\n5 88 0.0354 0.0157 -0.0026\n20 88 -0.1957 0.1996 0.0565\n50 88 -0.2838 0.2842 0.0518\n100 88 -0.4423 0.2965 0.0045\n200 88 -0.0758 -0.0067 0.017\n300 88 0.1725 0.1240 0.1792\n400 88 0.1782 0.1338 0.2131\n500 88 0.0687 0.2923 0.3047\n700 88 0.2157 -0.3502 -0.1325\n1000 88 0.3343 -0.3702 -0.0757\n1500 88 0.2065 -0.6317 -0.2844\n\nTable 5.9: The table of cross-talk estimators calculated using the equation\n5.58 by choosing randomly baseline positions b. All glitches g (130 glitches) of\nthe reference pixel 88 was taken into account. C1(b) is the cross-talk estimator\nis calculated for 8 pixels in round 1 (the green circle as shown in figure 5.47).\nSimilarly, C2(b) the cross-talk estimator is calculated for 12 pixels in round 2\n(the blue circle as shown in figure 5.47). Finally, C(b) the cross-talk estimator\n\nis calculated for whole 20 pixels.\n\nradioactive source position pixel 88. With very far baseline position b (big values\nof b), it means that the outside of the glitch window is the dominated noise signal.\nThe contribution of complex noise affects to give a constraint on the cross-talk\nestimator.\n\nSecondly, in order to observe the cross-talk estimator as a function of the number\nof glitches event g. We can consider this is the distribution of the cross-talk\nestimator respected to glitches. Figure 5.49 shows the distribution of cross-talk\nestimator C(b = 200, pixel area) and C(b = 300, pixel area) for the different\ndefined round of pixels for examples of the baseline position b = 200 and b = 300.\nThe distributions can be fitted by the Gaussian-like and zero-like center, those\nfitted values of mean µ and standard deviation σ are labeled in the figure. The\nnegative part means that the distribution of cross-talk estimators has contributed\nby noise.\n\nCross-talk of the long τ0 (the second population of the time constant τ0)\n\nAs discussed in the subsection 5.6.3, we also decided to study the level of cross-talk\nwhen a second population glitch (figure 5.36) occurs. For this reason, we select\nglitches with τ0 > 32ms and a validated fit (χ2 < 5.0). In order to obtain better\n\n\n\nCross-talk 207\n\n2 0 2 4\nCross-talk estimator\n\n0\n\n10\n\n20\n\n30\n\n40\n\n50\nGl\n\nitc\nh \n\nev\nen\n\nt\n1=-0.01 1=0.39\n2=-0.00 2=0.42\n=0.003 =0.72\n\nround1\nround2\nround\n\n2 0 2 4\nCross-talk estimator\n\n0\n\n10\n\n20\n\n30\n\n40\n\nGl\nitc\n\nh \nev\n\nen\nt\n\n1=0.01 1=0.42\n2=0.01 2=0.40\n=0.04 =0.73\n\nround1\nround2\nround\n\nFigure 5.49: The histograms of cross-talk estimator respects to the number\nof glitch event. The green, blue and the black curves represent for round 1 (the\ngreen circle - pixels: 69, 76, 75, 87, 34, 35, 93, 81), round 2 (the blue circle -\npixels: 128, 127, 74, 86, 33, 46, 47, 48, 41, 94, 82, 70), round square (whole\npixel of the green circle and the blue circle), respectively. The corresponding\nvalues of µ and σ of the fitted Gaussian are labeled in the plot. The x-axis shows\namplitudes of the cross-talk estimators while the y-axis indicates occurrence of\nglitch maximum. The baseline positions are 200 (left) and 300 (right) compared\n\nwith the maximum position.\n\nstatistical estimations of the cross-talk estimator (equation 5.58), a window of the\nbaseline position b from the glitch maximum position is considered.\n\nWhile a full glitch signal was defined as 750 bin samples. Each cross-talk estimator\nC(b) is calculated for each baseline position b in a chosen window, then the final\ncross-talk estimator C is evaluated by calculation the mean and the standard\ndeviation of all C(b) in that chosen window. With different windows of the baseline\nposition b, the table 5.10 shows the value of the mean and the standard deviation of\nthe cross-talk estimator in percentage for all pixels in the black square as illustrated\nin figure 5.47. The result shows that the cross-talk estimator depends on the\nconsidered window of the baseline position b and the value of cross-talk estimator\nis of the order of 0.1 %. When the window of the baseline position b is very large,\nit means that the outside a full glitch signal is noise signal, then these results give\nthe information of noise among pixels.\n\nOn the other hand, the contribution of noise plays an important role in the esti-\nmation of the thermal cross-talk estimator. The exactly same analysis method for\nthe data but the position of all pixels is shifted to the noise position n instead of\n\n\n\nCross-talk of the electronic readout chain 208\n\nWindow of Pixel µ% σ%\nbaseline b (C) (C)\n\n20 88 0.083 0.060\n50 88 0.1613 0.0830\n70 88 0.1934 0.0896\n100 88 0.1774 0.0802\n200 88 0.0107 0.1990\n300 88 -0.1010 0.2344\n500 88 -0.1452 0.2017\n750 88 -0.0389 0.259\n\nTable 5.10: The table of thermal cross-talk values calculated using the equation\n5.58 by choosing randomly windows of the baseline position b.\n\nthe position of glitch maximum i. The equation 5.58 become:\n\nCn(b, pixel area) =\n1\n\nNg\n\n1\n\nNj\n\nNg∑\n\ng=1\n\nNj∑\n\nj=1\n\ndj\ni(g)+n − 0.5dj\n\ni(g)+n−b − 0.5dj\ni(g)+n+b\n\nd0\ni(g)\n\n. (5.59)\n\nHere n is the noise position which is considered to outside the glitch window. The\nreference pixel is pixel 88. Because of fluctuation of noise then a window of the\nbaseline position b is also taking for the estimation of the cross-talk estimator\nof noise. The cross-talk estimator is calculated for each baseline position b in\na window then the final cross-talk estimator of noise is the mean and standard\ndeviation of those values. Comparing results in the table 5.11 and the table 5.10\n\nPosition n Window Pixel µn% σn%\nbaseline b (Cn) (Cn)\n\n1500 20 88 -0.0736 0.0559\n1500 50 88 -0.150 0.0760\n1500 70 88 -0.1746 0.0726\n1500 100 88 -0.1489 0.0769\n1500 200 88 -0.0087 0.27845\n1500 300 88 0.3460 0.3130\n1500 500 88 0.5077 0.3756\n1500 750 88 0.5934 0.3324\n\nTable 5.11: The table of contribution of noise level to thermal cross-talk esti-\nmator following the equation 5.59.\n\nwe observe that the behavior of noise performance in the signal is complex, it\naffects to the thermal cross-talk among pixels. As a conclusion, the thermal cross-\ntalk is constrained to less than 0.1 percent. The low statistic of events does not\nallow to put a better constraint.\n\n\n\nCross-talk of the electronic readout chain 209\n\n5.7.2 Cross-talk of the electronic readout chain\n\nAs described in section 5.3 (electronic readout system), we have been developed a\n128-to-1 time domain multiplexer (TDM) for the readout of the TES array. At the\nfirst multiplexing stage, each TES is readout by a SQUID. The 128 SQUIDs are\narranged in 32 rows and 4 columns as shown in figure 5.17. The Cbias capacitor\naimed to isolate each SQUID column to avoid cross-talk has a value of 100 nF\n[123]. At the second multiplexing stage, the ASIC provides sequentially readout\n4 columns multiplexing (\"Mux\") as shown in figure 5.50. The steps between two\nsuccessive pixels can introduce electronic cross-talk related to the frequency acqui-\nsition facq. Therefore, to do our measurements and to be able to analyze precisely\nthe glitches we use a fast sampling rate of 0.64 ms (1562.5 Hz). The fact is that\nQUBIC experiment has been first developed to use a lower frequency acquisition\n(∼ 156 Hz) for observing CMB photons, and to match a typical scanning strategy.\n\n1 33\n\n4 SQUIDs\n\nAS\nIC\n\n 1\n28\n\n:1\n65 97\n\n2\n\n11\n\n32\n\n24\n\n34\n\n43\n\n64\n\n56\n\n69\n\n75\n\n96\n\n88\n\n101\n\n107\n\n128\n\n120\n\n32\n S\n\nQ\nU\n\nID\ns\n\nMUX \n4:1\n\nFigure 5.50: The 32x4 SQUIDs and the ASIC multiplexing readout. Steps\nbetween two successive pixels can introduce cross-talk.\n\nFigure 5.51 shows the data of pixel 75 and pixel 107, these two successive pixels\nare sequential steps in multiplexing time, note that the pixel 75 is close to the\nradioactive source while the pixel 107 is far. The pixel 107 presents evidence of\nelectronic readout cross-talk at each glitch events. The perturbed signal between\nnegative and positive can be understood as resulting of Cbias by the changing sign\n\n\n\nCross-talk of the electronic readout chain 210\n\nFigure 5.51: cross-talk of the electronic readout chain between successive pix-\nels. The negative and positive fluctuated signal regime is probably due to the\nsuccessive positive and negative readout of the 4 SQUIDs columns used to charge\n\nand discharge capacitors Cbias.\n\n\n\nConclusion and discussion 211\n\nof the amplifiers. On the other hand, the plot also shows the group pixels 24, 56,\n88 (in front of radioactive source), 120, they are sequential in the readout timeline\nand the cross-talk is present at each glitch events. Let us consider three pixels\nin the sequential multiplexing timeline. The glitch event occurs at pixel 88, the\nelectronic readout cross-talk on pixel 120 is understandable in the readout timeline\nbecause pixel 120 is read after. Nevertheless, the signal appears on pixel 56 which\nis a past signal in the multiplexing timeline. Thus the interpretation of the cross-\ntalk of the electronic readout chain dose not work this case. We also consider the\ncase of particles which could hit directly the electronic readout chain, however the\ndistance of the SQUID stage and the radioactive source is very far and separated\nby the back copper of the TES array. This study needs more investigation to have\nfinal conclusions.\n\nFinally, there is many manners to study cross-talk than the thermal one or the one\nintrduced by multiplexing. The frequency acquisition of multiplexing can intro-\nduce one important cross-talk between two successive pixels (in the multiplexed\ntimeline). Indeed, it could also inductive between 2 SQUIDs nearby, or due to\nunmatched capacitors Cbias used to separate the SQUID biasing. The radioactive\nsource method allow the study of cross-talk, however this study requires a deeper\nwork. This thesis introduces the problem.\n\n5.8 Conclusion and discussion\n\nIn order to understand time constants of a TES detector, I studied the fundamental\ntheory of superconductivity, the principle of the electrical and thermal response of\na TES and its readout system. I developed a thermal model of TES to understand\nthe interaction of particles with a TES array. Thanks to presence of the cryostat\nequiped with the electronic feedback system including FLL, SQUID, ASIC, FPGA,\nPID, QUBIC studio . . . in the \"mm lab\" of APC laboratory, the data have been\ncollected and analyzed. The first step was to analyze TES I-V curves which\nrepresent the operating regime (the normal state, the superconducting transition,\nand the superconducting state) of a TES as a function of the voltage-bias in the\nelectrothermal feedback regime. We understand that we have 3 time constants\nin the system: (1) the intrinsic electric time constant of TES τel which is to\nsmall to be observable, (2) the electronic time constant of the readout chain, (3)\none thermal time constant describing the thermal behaviour of the TES. Time\n\n\n\nConclusion 212\n\nconstants of the TES depend on the biasing power applied to the TES and on the\nPID controller of the electronic feedback system. Those parameters are Vbias and\nKI(PID), respectively. In particular, if we apply the same power for TES (Vbias),\nwe increase the KI(PID) parameter, time constants will decrease. In conclusion,\nwe can choose an optimal parameter of KI for a QUBIC’s TES array based on\ntime constants behavior.\n\nIn a second part of this work, I studied cross-talk between TESs. Due to the fact\nthat the collected data of signal and noise have different behavior when we change\nthe frequency acquisition. This might have an effect on the time constant and\ncross-talk of the electronic readout chain. Indeed, two successive pixels introduced\nthe electronic cross-talk in the multiplexing timeline. Furthermore, the study of\ncross-talk needs more careful investigation especially the cross-talk of the electronic\nreadout system between two successive pixels which is necessary to improve for the\nQUBIC experiment. In addition, a discussion of time domain multiplexing and\nfrequency domain multiplexing (FDM) can be found in the paper [121]. Future\nstudies will rely on a study of the sampling rate and cosmic rays.\n\nOn the other hand, the TES array used for this data analysis is one of the QUBIC\nTES arrays. Due to the fact that the QUBIC technical demonstrator is developing\nsteadily after each fabricated TES array, many tests are carrying on and improving\neffectively the future TES arrays, they should have a better yield. Therefore, the\nsame study for the future TES arrays will have to be performed to get a better\nresult. For example, the material, Niobium compound, size grid are studied and\nevaluated carefully after each fabricated TES array.\n\nIn the aspect of technological discussion on the cross-talk, a QUBIC’s TES ar-\nray has different design as compared to a TES array of other experiments. For\nexample, POLARBEAR uses a sinuous antenna-coupled TES through bandpass\nfilters, the BICEP uses a planar antenna-coupled TES though bandpass filters.\nQUBIC has a dichroic mirror in which separates incident radiations into different\nfrequency bands directly to a TES array in the focal plane. As Expected, this\nprocedure somehow mitigates the cross-talk between frequency band level while\nfor the antenna coupled TES and bandpass filter, the interference might happen\nbetween different bands.\n\n\n\nChapter 6\n\nConclusion and perspectives\n\nIn this manuscript, I report the work I have been done during my Ph.D. I have\nfocused on systematic effect for future CMB satellite mission in a first part and\nthen I work on TES bolometer testing for the QUBIC experiment.\n\nThe first part is dedicated to bandpass mismatch systematic effect as one of the\nimportant systematic effects that can affect the current and next-generation mea-\nsurement of the polarization of the Cosmic Microwave Background radiation. The\nslightly different frequency bandpasses among detectors introduce leakage from\nintensity into CMB polarization. I have evaluated the level of the bandpass mis-\nmatch systematic effect for several observational strategies. The amplitude of\nthe leakage depends on the scanning strategy and impacts the estimation of the\ntensor-to-scalar ratio r. The result of the study allows us to optimize the scanning\nstrategy of future CMB projects. Particularly in case without a half-wave plate,\nwith the help of a full focal plane simulations at 140 GHz, random variation of\ndetector filters is of the order of 0.6 % I found that the spurious angular power\nspectrum could potentially bias on r at the reionization bump (` ≤ 10) at the level\n\nof 5 × 10−4 and the amplitude scale as\n1\n\nNdet\n\n. I have shown and verified the tight\n\ncorrelation between leakage maps and the average angle 〈cos 2ψ〉; 〈sin 2ψ〉. In case\nof an ideal continuously rotating HWP, the spectrum of the bandpass mismatch\nerror leakage is similar to white noise. In order to obtain accuracy evaluations on\nthe bandpass mismatch error, we have to concern to precise assumptions on (1)\nthe scanning pattern strategy, (2) the variations in the bandpass filters, and (3)\nforeground removal process, (4) frequency band (5) 1/f noise in the modeling.\n\n213\n\n\n\nConclusion 214\n\nIn the second part of the manuscript, in order to study the interaction of cosmic\nrays with the focal plane unit of future CMB projects, I studied the behavior of\nTES arrays of the ground-based QUBIC experiment in the laboratory. We have\nmounted on the front of a TES array an 241Am source extracted from common\nsmoke detectors. This source is particularly suitable for our tests as its main\ndecay products are alpha particles with 5.5 MeV energy. Our analysis of the TES\nresponse to the glitch interaction, the study indicates that (1) we have clearly\nmeasured two time constants, the rising time constant τ0, which is mostly due\nto the system limited bandwidth of the electronic readout chain and the thermal\ntime constant (τ1), which represents decay time of a glitch. Typical values for τ0\n\nand τ1 are (7-30) ms and (20-60) ms, respectively. Moreover, we estimated the\nenergy of the alpha particles in the range (2-5) MeV, in agreement with theoretical\npredictions. Our results demonstrate the capability of our readout electronics in\nperforming tests of sensitivity towards CRs. (2) The rising time constant (τ0)\nof the pixel 88 (in front of the radioactive source) has two populations, the first\npopulation can be interpreted as particles hit directly to the thermometer or the\nabsorber of the sensor, the thermal equilibrium is rapidly established due to the\ndeposited energy on the 1 µm thickness of the absorber. The second population\ncan be interpreted as particles hit the Si substrate which has the thickness of 500\nµm, so that the deposited energy is huge. The heat flows increase the background\nreference temperature which affects the sensor through the thermal link between\nthe sensor and the Si substrate. The sensor finally detected through a rising time\nτ0 dominated by the thermal time constant (more or less equal to τ1). The result of\nother pixels have difficulties to observe clearly the second population of the rising\ntime τ0 of the electronic readout chain. Several factors could affect the result such\nas the position of the pixel compared to the radioactive source, the thermal cross-\ntalk among pixels, the behavior of the electronic readout system. (3) The thermal\ncross-talk has been estimated using an estimator. On the other hand, I found\nthat the frequency acquisition of the multiplexing readout chain can introduce the\ncross-talk between two successive pixels in the multiplexing timeline. This could\nbe a problem for the electronic readout system, however we studied interactions\nof α particles (fast signal compare to CMB) with the TES array using a very\nhigh frequency acquisition (1562.5 Hz). In fact QUBIC experiment has been first\ndeveloped for a lower frequency acquisition (∼ 156 Hz). Furthermore the study\nof the cross-talk of the electronic readout system needs a deeper work, this is a\nprospective of the thesis.\n\n\n\nConclusion 215\n\nIn this manuscript, I described the chapter 2 introduction to modern cosmology,\nstandard cosmological model and the chapter 3 description of CMB in spherical\nharmonic, temperature anisotropies, CMB polarization, foreground components,\nand systematic effects. I am happy to state that there have plenty of avenues for\nimproving and extending the work after my Ph.D.\n\nSystematic effects\n\nI have studied bandpass mismatch systematic effect, with similar procedures I\ncould also study others systematic effects as the gain mismatch, cosmic rays effect,\n1/f noise performance as well as beam asymmetry . . . The study of imperfection in\nfrequency bands of a half-wave plate is also an interesting topic close to bandpass\nmismatch.\n\nForeground components separation\n\nA perspective of the bandpass mismatch study is that we need to study accuracy re-\nquirements of bandpass filters variation for future CMB experiment. Furthermore,\nthe bandpass mismatch among detectors will have an impact to the component\nseparation of the foreground. In fact that the bandpass mismatch study is achieved\nat 140 GHz dominated by thermal dust and synchrotron. If the foregrounds turn\nout to be complicated, we might have large bandpass errors thus a foreground\ncleaning and calibration method are necessary. In addition, the study of bandpass\nmismatch effect for multi-channel frequency bands also can be a future interesting\ntopic.\n\nSensor Technology\n\nIn this manuscript especially chapter 5, I described the principle of TES as well\nas the behavior of a TES array to particle interactions. One thing missing is\nthe TES design architecture and fabrication which should be pursued to cope\nthe problem which has been identified and it is a straightforward direction in my\nresearch career. The fact is that the fabrication processes of an array are difficult,\nspecially suspended membranes. It is also mandatory to have a good control over\nfilm and layers. The readout system of TES is very complex, SQUIDs are tricky\nand extremely expensive, difficult to multiplex and readout. Therefore I have put\nmy eyes on the perspective of Kinetic Inductance Detector (KID) which has a\nsimple readout system that uses a resonance circuit to see resonance frequency\nchange when photons strike.\n\n\n\nAppendix A\n\nSolutions of Einstein equations\n\nA.1 FLRW solution and Friedmann equations\n\nThe FLRW metric is:\n\nds2 = c2dt2 − a2(t)\n\n(\ndr2\n\n1− kr2\n+ r2\n\n(\ndθ2 + sin2 θdϕ2\n\n))\n.\n\nThe diagonal metric coefficients are\n\ng00 = 1;\n\ng11 = − a2(t)\n\n1− kr2\n;\n\ng22 = −a2(t)r2;\n\ng33 = −a2(t)r2 sin2 θ;\n\ng00 =\n1\n\ng00\n\n= 1;\n\ng11 =\n1\n\ng11\n\n= −1− kr2\n\na2(t)\n;\n\ng22 =\n1\n\ng22\n\n= − 1\n\na2(t)r2\n;\n\ng33 =\n1\n\ng33\n\n= − 1\n\na2(t)r2 sin2 θ\n;\n\ngνβgαβ = δνα; gµν = gµαgνβgαβ; Aµ = gµνA\nν ; Aµ = gµνAν .\n\n216\n\n\n\nAppendix A. Solutions of Einstein equations 217\n\nA.1.1 Christoffel symbols for the FLRW metric\n\nΓµαβ ≡\ngµν\n\n2\n[gαν,β + gβν,α − gαβ,ν ] =\n\ngµν\n\n2\n\n[\n∂gαν\n∂xβ\n\n+\n∂gβν\n∂xα\n\n− ∂gαβ\n∂xν\n\n]\n.\n\nΓνµρ =\n1\n\n2\n[gµν,ρ + gνρ,µ − gρµ,ν ] .\n\nΓ0\n00 = 0\n\nΓ0\n0i = Γ0\n\ni0 = 0\n\nΓµαν = Γµνα\n\nThe non-zero Christoffel symbols are:\n\nΓ0\n11 =\n\n1\n\nc\n\naȧ\n\n1− kr2\n; Γ0\n\n22 =\n1\n\nc\naȧr2; Γ0\n\n33 =\n1\n\nc\naȧr2 sin2 θ;\n\nΓ1\n01 = Γ1\n\n10 =\n1\n\nc\n\nȧ\n\na\n; Γ1\n\n11 =\nkr\n\n1− kr2\n; Γ1\n\n22 = −r\n(\n1− kr2\n\n)\n; Γ1\n\n33 = −r\n(\n1− kr2\n\n)\nsin2 θ;\n\nΓ2\n02 = Γ2\n\n20 = Γ3\n03 = Γ3\n\n30 =\n1\n\nc\n\nȧ\n\na\n; Γ2\n\n12 = Γ2\n21 = Γ3\n\n13 = Γ3\n31 =\n\n1\n\nr\n;\n\nΓ2\n33 = − sin θ cos θ; Γ3\n\n23 = Γ3\n32 =\n\ncos θ\n\nsin θ\n;\n\nA.1.2 Ricci tensor and Einstein’s tensor\n\nThe Ricci tensor is:\n\nRµν = Γαµν,α − Γαµα,ν + ΓαβαΓβµν − ΓαβνΓ\nβ\nµα,\n\nR00 = − 3\n\nc2\n\nä\n\na\nc=1\n=\n\n3ä\n\na\n;\n\nR11 =\n1\n\n1− kr2\n\n(\näa\n\nc2\n+\n\n2ȧ2\n\nc2\n+ 2k\n\n)\nc=1\n=\n\naä+ 2ȧ2 + 2k\n\n1− kr2\n;\n\n\n\nAppendix A. Solutions of Einstein equations 218\n\nR22 = r2\n\n(\näa\n\nc2\n+\n\n2ȧ2\n\nc2\n+ 2k\n\n)\nc=1\n= r2\n\n(\naä+ 2ȧ2 + 2k\n\n)\n;\n\nR33 = r2 sin2 θ\n\n(\näa\n\nc2\n+\n\n2ȧ2\n\nc2\n+ 2k\n\n)\nc=1\n= r2 sin2 θ\n\n(\naä+ 2ȧ2 + 2k\n\n)\n;\n\nRicci Scalar is:\n\nR ≡ gµνRµν = Rα\nα = −6\n\n[\nä\n\na\n\n1\n\nc2\n+\nȧ2\n\na2\n\n1\n\nc2\n+\nk\n\na2\n\n]\nc=1\n= −6\n\naä+ ȧ2 + k\n\na2\n;\n\nEinstein’s tensor is:\n\nG00 = 3\nȧ2\n\na2\n\n1\n\nc2\n+ 3\n\nk\n\na2\n;\n\nGii = gii\n\n[\n2\nä\n\na\n\n1\n\nc2\n+\nȧ2\n\na2\n\n1\n\nc2\n+\nk\n\na2\n\n]\n;\n\nA.2 Stress-energy tensor Tµν\n\nT µν = Tµν =\n(\nρc2 + P\n\n) uµuν\nc2\n− Pgµν ,\n\nThe stress-energy tensor components are:\n\nTii = Pgµν ;\n\nT00 = (ρ+ P )\nu0u0\n\nc2\n− Pg00 = ρ;\n\nT11 = P\na2\n\n1− kr2\n;\n\nT22 = Pr2a2;\n\nT33 = Pa2r2 sin2 θ;\n\nUsing those calculations above, we could find the Friedmann equations.\n\n\n\nAppendix A. Solutions of Einstein equations 219\n\nA.3 Schwarzschild Solution and Black Holes\n\nThe Schwarzschild metric so-called Schwarzschild solution of Einstein equations is\n\nds2 = −\n(\n\n1− 2GM\n\nc2r\n\n)\ncdt2 +\n\n(\n1− 2GM\n\nc2r\n\n)−1\n\ndr2 + r2\n(\ndθ2 + sin2 θdϕ2\n\n)\n.\n\nWhere G is the universal gravitational constant, M is a Newtonian mass. The\nequation is the static spherically symmetric vacuum solution of Einstein equations.\nThe equation describes space-time around a point mass. There are two interesting\npoints at r = 0 which represents a real singularity and at the Schwarzschild radius\nrS =\n\n2GM\n\nc2\n, the radius at which mass M collapses into a black hole.\n\n\n\nAppendix B\n\nχ2 and fit C`\n\nIf we have an equation d = ax + n, then n = d − ax, where d is the data mea-\nsurement, x is a model of data, a is an estimate solution and n is the noise\ncontribution. The probability distribution function of noise is assumed as a Gaus-\nsian distribution: P (n) = 1\n\nN(σ)\ne−n\n\ntN−1n\n\nχ2 = − log(P (n)), maximum P (n) corresponds to minimum χ2\n\nχ2 = ntN−1n =\n∑\n\n(di − axi)\n1\n\nσ2\ni\n\n∑\n(di − axi) =\n\n∑\n(di − axi)2\n\nσ2\ni\n\n.\n\nMinimum χ2, it means that\n\n∂χ2\n\n∂x|a\n=\n−2\n\nσ2\ni\n\n∑\nxi(di − axi) = 0.\n\n∑\nxidi\nσ2\ni\n\n=\n\n∑\naxi\n\n2\n\nσ2\ni\n\n,\n\na =\n\n∑\nxidi\nσ2\ni∑\nxi2\n\nσ2\ni\n\n,\n\nWe can imply the form of the solution:\n\na =\n∑(\n\nxi\n1\n\nσ2\ni\n\nxi\n\n)−1∑\nxi\n\n1\n\nσ2\ni\n\ndi,\n\n220\n\n\n\nAppendix C. χ2 and fit C` 221\n\nWe can apply the χ2 method to fit the cosmic variance C` to estimate the tensor\nto scalar value r\n\nχ2 =\n∑\n\n`\n\n(\nC̃` − 100rC`0.01\n\n)2\n\n2C2\n`\n\n2`+1\n+\n\nN2\n`\n\n2`+1\n\n,\n\nthen we apply the equation above by consider noise amplitude N` is >> than C`,\nand C̃`, C` are the data measurement and the model respectively.\n\n100r =\n∑\n\n`\n\n(\nC2\n`\n\n2`+ 1\n\nN2\n`\n\n)−1∑\n\n`\n\n(\nC`C̃`\n\n2`+ 1\n\nN2\n`\n\n)\n,\n\n100r =\n∑\n\n`\n\n(\nC2\n` (2`+ 1)\n\n)−1\n∑\n\n`\n\n(\nC`C̃`(2`+ 1)\n\n)\n.\n\n\n\nAppendix C\n\nFitted glitches\n\nIn this appendix, I present the fitted glitches for the data run7 and the pixel 88\nwhich is in front of the radioactive source. 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Sekimoto et al., Concept design of the LiteBIRD\nsatellite for CMB B-mode polarization, SPIE Astronomical Telescopes + Instru-\nmentation, 2018, Texas-USA, DOI: 10.1117/12.2313432\n\n[5] XIIIth School of Cosmology, The CMB from A to Z, Cargès: Duc Thuong\nHoang, Optimization of the next generation of CMB missions [talk]\n\n[6] 53rd Rencontres de Moriond: Duc Thuong Hoang, Evaluating the level of the\nbandpass mismatch systematic effect for the future CMB satellites. [proceeding]\n\n[7] 53rd Rencontres de Moriond: QUBIC collaboration, G. D’Alessandro et al.,\nThe QUBIC experiment [proceeding]\n\n[8] QUBIC collaboration, M. Salatino, Benoit Bélier , Claude Chapron , Duc\nThuong Hoang et al., Performance of NbSi Transition-Edge Sensors read out with\na 128 MUX factor for the QUBIC experiment , SPIE Astronomical Telescopes +\nInstrumentation, 2018, Texas-USA, DOI: 10.1117/12.2312080\n\n[9] QUBIC collaboration, D. Burke et al., Optical modelling and analysis of the\nQ and U bolometric interferometer for cosmology, International Society for Optics\nand Photonics, 2018, California-USA DOI: 10.1117/12.2287158\n\n[10] QUBIC collaboration, C. O’Sullivan et al., QUBIC: the Q and U bolometric\ninterferometer for cosmology, SPIE Astronomical Telescopes + Instrumentation,\n2018, Texas-USA, DOI: 10.1117/12.2313332\n\nhttp://iopscience.iop.org/article/10.1088/1475-7516/2017/12/015\nhttp://iopscience.iop.org/article/10.1088/1475-7516/2018/04/022/meta\nhttp://iopscience.iop.org/article/10.1088/1475-7516/2018/04/014/meta\nhttps://doi.org/10.1117/12.2313432\nhttp://www.cpt.univ-mrs.fr/~cosmo/EC2017/Presentations/Seminaires/Hoang.pdf\nhttps://doi.org/10.1117/12.2312080\nhttps://doi.org/10.1117/12.2287158\nhttps://doi.org/10.1117/12.2313332\n\n\nPublications and scientific activities 246\n\n[11] QUBIC collaboration, A. J. May et al., Thermal architecture for the QUBIC\ncryogenic receiver, SPIE Astronomical Telescopes + Instrumentation, 2018, Texas-\nUSA, DOI: 10.1117/12.2312085\n\n[12] QUBIC collaboration, C. O’Sullivan et al., Simulations and performance of the\nQUBIC optical beam combiner, SPIE Astronomical Telescopes + Instrumentation,\n2018, Texas-USA, DOI: 10.1117/12.2313256\n\n[13] QUBIC collaboration, P de Bernardis et al., QUBIC: Measuring CMB po-\nlarization from Argentina, Boletin de la Asociacion Argentina de Astronomia La\nPlata Argentina, Vol, 60\n\n[14] QUBIC collaboration, Aniello Mennella et al., QUBIC: Exploring the primor-\ndial Universe with the Q&U Bolometric Interferometer\n\n[15] Ranajoy Banerji, Jacques Delabrouille, Guillaume Patanchon, Duc Thuong\nHoang, Martin Bucher, Tomotake Matsumura, Hirokazu Ishino, Masashi Hazumi,\nBandpass mismatch error for satellite CMB experiments II: Correcting for the\nspurious signal. [in preparation]\n\n[16] My initiative : 1st Meeting of Young Vietnamese Community of Astronomy\n(YVCA), 21-22 December 2017, APC laboratory, Université Paris Diderot, Paris-\nFrance, program: https://space.usth.edu.vn/en/news/news-events/yvca-program-\n127.html\n\n https://doi.org/10.1117/12.2312085\nhttps://doi.org/10.1117/12.2313256\nhttp://www.astronomiaargentina.org.ar/b60/2018baaa...60...107B.pdf\nhttps://space.usth.edu.vn/en/news/news-events/yvca-program-127.html\nhttps://space.usth.edu.vn/en/news/news-events/yvca-program-127.html\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nournal of Cosmology and Astroparticle Physics\nAn IOP and SISSA journalJ\n\nBandpass mismatch error for satellite\nCMB experiments I: estimating the\nspurious signal\n\nDuc Thuong Hoang,a,b Guillaume Patanchon,a Martin Bucher,a,c\n\nTomotake Matsumura,d,e Ranajoy Banerji,a Hirokazu Ishino,f\n\nMasashi Hazumig,e,d,h and Jacques Delabrouillea,i\n\naLaboratoire Astroparticule et Cosmologie (APC), Université Paris Diderot, CNRS/IN2P3,\nCEA/Irfu, Observatoire de Paris, Sorbonne Paris Cité,\n10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France\nbDepartment of Space and Aeronautics, University of Science and Technology of Hanoi\n(USTH), Vietnam Academy of Science and Technology (VAST),\n18 Hoang Quoc Viet, Cau Giay District, Hanoi, Vietnam\ncAstrophysics and Cosmology Research Unit,\nSchool of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal,\nDurban 4041, South Africa\ndKavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI),\nUTIAS, The University of Tokyo,\nKashiwa, Chiba 277-8583, Japan\neInstitute of Space and Astronautical Science (ISAS),\nJapan Aerospace Exploration Agency (JAXA),\nSagamihara, Kanagawa 252-0222, Japan\nfDepartment of Physics, Okayama University,\n3-1-1 Tsushimanaka, Kita-ku, Okayama 700-8530, Japan\ngHigh Energy Accelerator Research Organization (KEK),\nTsukuba, Ibaraki 305-0801, Japan\nhThe Graduate University for Advanced Studies (SOKENDAI),\nMiura District, Kanagawa 240-0115, Hayama, Japan\niDépartement d’Astrophysique, CEA Saclay DSM/Irfu,\n91191 Gif-sur-Yvette, France\n\nE-mail: hoang@apc.in2p3.fr, guillaume.patanchon@apc.univ-paris-diderot.fr,\nbucher@apc.univ-paris7.fr, tomotake.matsumura@ipmu.jp, banerji@apc.in2p3.fr,\nscishino@s.okayama-u.ac.jp, masashi.hazumi@kek.jp, delabrou@apc.in2p3.fr\n\nReceived July 4, 2017\nRevised November 15, 2017\nAccepted November 18, 2017\nPublished December 7, 2017\n\nc© 2017 IOP Publishing Ltd and Sissa Medialab https://doi.org/10.1088/1475-7516/2017/12/015\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nAbstract. Future Cosmic Microwave Background (CMB) satellite missions aim to use the\nB mode polarization to measure the tensor-to-scalar ratio r with a sensitivity σr <∼ 10−3.\nAchieving this goal will not only require sufficient detector array sensitivity but also unprece-\ndented control of all systematic errors inherent in CMB polarization measurements. Since\npolarization measurements derive from differences between observations at different times\nand from different sensors, detector response mismatches introduce leakages from intensity\nto polarization and thus lead to a spurious B mode signal. Because the expected primordial B\nmode polarization signal is dwarfed by the known unpolarized intensity signal, such leakages\ncould contribute substantially to the final error budget for measuring r. Using simulations\nwe estimate the magnitude and angular spectrum of the spurious B mode signal resulting\nfrom bandpass mismatch between different detectors. It is assumed here that the detectors\nare calibrated, for example using the CMB dipole, so that their sensitivity to the primor-\ndial CMB signal has been perfectly matched. Consequently the mismatch in the frequency\nbandpass shape between detectors introduces differences in the relative calibration of galac-\ntic emission components. We simulate this effect using a range of scanning patterns being\nconsidered for future satellite missions. We find that the spurious contribution to r from the\nreionization bump on large angular scales (` < 10) is ≈ 10−3 assuming large detector arrays\nand 20 percent of the sky masked. We show how the amplitude of the leakage depends on\nthe nonuniformity of the angular coverage in each pixel that results from the scan pattern.\n\nKeywords: CMBR experiments, CMBR polarisation\n\nArXiv ePrint: 1706.09486\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nContents\n\n1 Introduction 1\n\n2 Sky emission model and mismatch errors 3\n\n3 Calculating the bandpass mismatch 5\n\n3.1 Results 8\n\n3.2 Analytic estimates 14\n\n3.3 Importance of avoiding resonances 18\n\n3.4 Hitcount and crossing moment map properties 20\n\n4 Conclusions 24\n\n1 Introduction\n\nMeasurements of the cosmic microwave background (CMB) provide a rich data set for study-\ning cosmology and astrophysics and for placing stringent constraints on cosmological models.\nIn particular, the ESA Planck satellite mission has produced full sky maps in both temper-\nature and polarization at unprecedented sensitivity in nine broad (∆ν/ν ≈ 0.3) microwave\nfrequency bands [1].\n\nConventional cosmological models predict that the CMB is linearly polarized, so that\nthe fourth Stokes parameter V vanishes. CMB polarization patterns can be decomposed\nin two components known as the E and B modes, respectively of even and odd parity. In\nlinear cosmological perturbation theory, scalar perturbations produce E mode polarization\nbut are unable to produce any B mode polarization at linear order. The E mode polarization\nangular power spectrum can be predicted from a model fitted to the measured T anisotropies.\nThe WMAP [2] and Planck [3] space missions, complemented on smaller angular scales by\nACT [4] and SPT [5], have already measured the E mode polarization power spectrum up to\nhigh multipole number `, even if the accuracy of the measurement can still be substantially\nimproved. On the other hand, the odd parity (or pseudo-scalar) polarization pattern called\nthe B mode arises either from primordial tensor perturbations, or equivalently primordial\ngravitational waves, presumably generated during inflation, or from scalar modes at higher\nnonlinear order, primarily through gravitational lensing. Gravitational lensing B modes\ndominate over primordial B modes on small angular scales. These gravitational lensing B\nmodes have already been observed at ` >∼ 100 by the POLARBEAR [6], SPT-Pol [7] and\nBicep2/Keck [8] ground-based experiments. Primordial B modes have not been observed\nyet. Their predicted shape features a ‘recombination bump’ visible around ` ≈ 80, and a\n‘reionization bump’ at ` <∼ 10. The overall amplitude of this primordial B mode spectrum\ndepends linearly on the value of the tensor-to-scalar ratio r. The current upper limit is\nr < 0.07 at 95% c.l. [9, 10].\n\nAfter Planck, a number of ground-based and balloon-borne experiments currently either\ntaking data or in the planning stage aim to make the first detection of primordial B modes. In\nparallel, the space-borne mission concepts CORE [11], LiteBIRD [12, 13], and PIXIE [14] have\n\n– 1 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nbeen designed to probe B modes at higher sensitivities and using frequency bands inaccessible\nfrom the ground. Constraining physically motivated inflation models requires sensitivities in\nthe tensor-to-scalar ratio of σr <∼ 10−3, almost two orders of magnitude beyond the Planck\nsensitivity. Furthermore, systematic errors must be controlled so that their contribution to\nthe final error budget is subdominant. The calibration requirements become correspondingly\nmore stringent, and future experiments will have to devise novel calibration procedures to\ncharacterize the instrument at a level that makes it possible to correct the raw data at\nsufficient accuracy.\n\nTypically experiments observe in a number of different frequency channels with many\ndetectors for each frequency channel. Ideally, all detectors in a single channel should have\nthe exact same bandpass function (i.e., the response g(ν) that defines the transmission of the\nsystem as a function of frequency) in order to construct single band maps, which are then\nanalyzed to isolate the primordial cosmological signal. Many detectors are necessary in each\nchannel to improve on the sensitivity of the current observations, which already use detectors\nthat are very nearly at the quantum noise limit. If however the detectors that are meant to\nbe identical have slightly different bandpasses, artifacts are introduced into the maps that\nare obtained by combining the signals from several detectors. After cross-calibration on\nthe CMB, for instance using the bright CMB dipole, the amplitude of other astrophysical\ncomponents is different in the different detectors, and residuals of the differences of integrated\nintensity leak into the reconstructed polarization maps. Such effects have been observed in\nPlanck [15] and WMAP [16]. In this paper we call these artifacts ‘bandpass mismatch errors’.\n\nObviously, such errors can be avoided if the observing strategy allows first to make\npolarization maps with each detector independently, hence without bandpass mismatch er-\nrors, and then to combine these individual detector maps into a global map. This however\nrequires observing each sky pixel with enough independent orientations of the detector polar-\nizer. This polarization modulation can be achieved either with the use of a rotating half-wave\nplate (HWP), or by rotating the whole instrument so that each pixel is observed with an\noptimized set of detector orientations. However, practical considerations may constrain the\nrange of possible polarization orientations, leading to a loss of sensitivity after combining\nsingle detector polarization maps.\n\nThe objective of this paper is to evaluate the level of the bandpass mismatch effect for\nfuture space missions and to estimate its possible impact on the final determination of the\ntensor-to-scalar ratio r if no correction measures are taken. Our study first focuses on the case\nwithout a HWP, and we also verify that the effect is greatly reduced with an ideal rotating\nHWP without any chromaticity or other non-idealities. For a more detailed discussion of\ngeneral issues pertaining to the use of a HWP for achieving polarization modulation and in\nparticular a discussion of the issue of achromaticity, we refer the reader to the results of the\nABS experiment [17] and the thesis [18] and references therein. We note that in the first case,\nmaking single detector maps that are subsequently combined to avoid bandpass mismatch\nerrors, could be done at the price of increased final noise since the angular coverage in each\npixel is sub-optimal. HWP non-idealities are not studied in this paper. Some of the issues\nconsidered in this paper are also discussed in ref. [19].\n\nIn section 2 we model the bandpass mismatch effect, and in section 3 we evaluate the\nimpact on B mode measurements and relate the mismatch errors to the “crossing moment\nmaps”, that provide a measure of uniformity of polarizer angle coverage in each pixel. Cor-\nrection methods are developed in a companion publication [20].\n\n– 2 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\n2 Sky emission model and mismatch errors\n\nThe total intensity of the microwave sky can be expressed as a sum of components of different\nastrophysical origin. In intensity, the CMB anisotropies are dominant over most of the sky,\nbut several diffuse components of galactic origin are also present as well as compact sources,\nwhich include extragalactic radio sources, IR sources (understood to be dusty galaxies), and\nSunyaev-Zeldovich (SZ) distortions from the hot gas within galaxy clusters. We model the\nunpolarized sky emission at position p̂ and frequency ν as\n\nItot(p̂, ν) = I0(ν) +\n∂B(ν;T )\n\n∂T\n\n∣∣∣∣∣\nT0\n\n∆TCMB(p̂) +\n∑\n\n(c)\n\nI(c)(p̂, ν) (2.1)\n\nwhere B(ν;T ) is the spectrum of a blackbody at temperature T , T0 is the average CMB\ntemperature of about 2.7255 K, ∆TCMB(p̂) is the CMB temperature fluctuation around this\nmean value, I(c)(p̂, ν) the emission spectrum of component (c) as a function of electromagnetic\nfrequency ν, I0(ν) is the monopole including all components. We have similar relationships\nfor the Q and U Stokes parameters. All three Stokes parameters of the CMB at a given\nposition on the celestial sphere have the factorized frequency dependence as given above.\nA similar factorizable form can be used for the SZ emission assuming that the hot gas is\nnon-relativistic. The galactic components are more complicated at the accuracy required for\nfuture satellite missions, and an Ansatz where the frequency dependence of each component\nfactorizes out breaks down. However for studying bandpass mismatch error to first order, a\nsimple factorizable model suffices.\n\nFor this bandpass mismatch study, we consider only the CMB and the diffuse galac-\ntic components, which contribute the largest bandpass mismatch error. At frequencies\n≈ 150 GHz where we focus our study, the galactic emission can be decomposed into thermal\ndust emission, which is the dominant component, and synchrotron, free-free, and spinning\ndust emissions. The carbon monoxide (CO) rotational emission at transition line frequencies\nν = 115 GHz for J = 1 → 0 and ν = 230 GHz for J = 2 → 1 was a source of significant\nleakage in Planck experiment [21], but is not considered here because we anticipate that the\nfilters used by future satellite experiments will avoid these lines.\n\nFor our study we assume that the galactic thermal dust emission is a greybody of\ntemperature Td ≈ 19.7 K [22] with an emissivity spectral index β(p̂), which depends on sky\nposition and whose average value is ≈ 1.62 as measured by Planck [22, 23]. The synchrotron\nand free-free emissions can be described by power law spectra with the negative spectral\nindices ≈ −3.1 and ≈ −2.3, respectively (see [24] and references therein). The fluctuation of\nthe signal (relatively to the average CMB monopole) measured by the detector i is given by\n\n∫\ndν gi(ν)\n\n(\nI(p̂, ν)− I0(ν)\n\n)\n=\n\n∫\ndν gi(ν)\n\n∂B(ν;T )\n\n∂T\n\n∣∣∣∣\nT0\n\n∆TCMB(p̂)\n\n+\n\n∫\ndν gi(ν) Id(p̂, ν0)\n\n(\nν\n\nν0\n\n)β(p̂) B(ν;Td)\n\nB(ν0;Td)\n+ . . . , (2.2)\n\nwhere I0(ν) = B(ν;T0) is the CMB monopole, gi(ν) is the bandpass function of the detector\ni, Id(p̂, ν0) is the amplitude of the dust component at the reference frequency ν0, and where\nthe dots stand for other components (such as synchrotron and free-free) not explicitly written\nhere. To first order we obtain for the total sky intensity Isky(ν0) after converting the CMB\n\n– 3 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\ntemperature ∆TCMB to intensity ICMB(ν0):\n\nIsky(ν0) = ICMB(ν0) + γd Idust(ν0) + γs Isync(ν0) + . . . , (2.3)\n\nwhere\n\nγd =\n\n\n\n\n∫\ndν gi(ν)\n\n(\nν\nν0\n\n)β\nB(ν;Td)\nB(ν0;Td)\n\n∫\ndνgi(ν)\n\n(\n∂B(ν;T )\n∂T\n\n) ∣∣∣\nT0\n\n\n\n(\n∂B(ν0;T )\n\n∂T\n\n) ∣∣∣∣\nT0\n\n. (2.4)\n\nThe factor γs is similarly defined integrating over the synchrotron spectrum, etc.\nEq. (2.3) also holds for the polarization when I is replaced with Q and U. The unit\n\nnormalization for the CMB component is justified because the data are calibrated using the\nCMB dipole (or higher order temperature anisotropies). The values of the γ parameters are\nclose to unity when the bandwidth is narrow.\n\nDifferences in the bandpass function gi(ν) from detector to detector result in correspond-\ning variations in γ from detector to detector for each non-CMB component. Such variations\nhave been observed in the Planck data (see figures 5 and 28 of [25] for the measured Planck\nfilters and the mismatch parameters, respectively). Pre-flight Fourier Transform Spectrome-\nter (FTS) ground measurements characterized variations of the filter edge positions at both\nthe low and high frequencies at about the percent level. Ground measurements, however,\nwere not accurate enough to detect variations near the center of the filters, and thus could not\nbe used to determine the γ parameters with sufficient accuracy. The γ parameter variations\nhad to be determined from the flight data to allow an accurate correction of the leakage (see\nthe low-` Planck paper [15] as well as [26]). It should be noted that the variations of the\nbandpass functions of the filters from detector to detector for a future satellite experiment\nwill depend on the kind of detector technology used (see also [27] regarding the WMAP\nexperiment).\n\nAs already stressed, for the above sky emission model where each component has a\nfixed (factorizable) frequency dependence, the bandpass mismatch maps depend only on the\nγ parameters and not on other details of the filters. The deviations from this simplified model\ndue to the observed spatial variations of the spectral indices of component spectra and of\nthermal dust temperature produce a second order correction to the bandpass mismatch error,\nwhich is neglected for this study. Consequently, the intensity to polarization leakage due to\nbandpass mismatch can be obtained using only the γ’s and no additional properties of the\nbandpass functions.\n\nTo relate these variations to filter properties, we assume a simple model in which each\nfrequency band is a tophat bandpass function for which g(ν) = 1 in the interval [νmin, νmax]\nand g(ν) = 0 elsewhere. We assume that the variations in νmin and νmax for each detector\nare generated independently according to a uniform distribution with a width of 1%.1\n\nWe also assume a bandwidth (νmax− νmin)/ν0 of 0.25 on average, with ν0 = 140.7 GHz.\nThe resulting RMS of γd is 0.6% for this simple model. This is similar to the variations\nobserved for Planck at 143 GHz. The fact that actual bandpass functions are more complex\n\n1We thank Aritoki Suzuki for sharing with us that the measurement errors with FTS in the bandpass of\nthe third-order Chebyshev filter placed between the broadband sinuous antennas and the bolometers of the\nfocal plane panels of the Simons Array [28] give approximately this spread. Obviously, since these are values\ndominated by measurement error, the actual bandpass mismatch for these filters could be much smaller. These\nmeasurements merely serve to establish an upper bound on the mismatch. These values are also of the same\norder of magnitude as the values representing the bandpass mismatch of the metal mesh filters used as part\nof the Planck satellite HFI instrument. [See [29] for a discussion of the Planck bandpass mismatch.]\n\n– 4 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nfunctions of ν does not affect the applicability of the present work as long as the corresponding\nγ coefficients remain of the same order of magnitude. Results for other values may be obtained\nby trivial rescaling. We verified the expected linear scaling by increasing the width of the\nuniform distribution from 1% to 2% and observed that the leakage increases by a factor of\n2, as expected.\n\nThis simple model for detector bandpasses is appropriate for the foreground components\nhaving a smooth frequency spectral dependence (e.g., synchrotron and dust emission), but for\ngalactic line emission (such as galactic CO emission and other spectral lines) a more detailed\nmodel would be required. The γ’s are computed as a random set from this distribution\nmodel, since those are the only quantities needed for the bandpass mismatch evaluation.\n\nIn this paper, we focus our analysis on a frequency channel centered at ν0 = 140 GHz,\nand so we restrict ourselves to the dominant galactic component, namely the thermal dust\nemission. More galactic components are included in the companion paper discussing the\ncorrection of the mismatch [20].\n\n3 Calculating the bandpass mismatch\n\nIn this section we use a simplified model of the measurement, stripped of additional com-\nplications such as asymmetric beams, pixelization effects, etc. for estimating the dominant\ncontribution to the bandpass mismatch error. A study of more than one source of systematic\nerrors simultaneously would obviously be more complicated and also less intuitive to inter-\npret. Here our purpose is to study bandpass mismatch error in isolation and in the simplest\npossible context.\n\nWe assume a scanning pattern that combines three rotations: a relatively fast spin of\nthe payload around a spin axis that precesses around the anti-solar direction, which itself\nfollows the yearly motion of the spacecraft around the Sun. Many of the proposed future\nCMB polarization space missions have adopted such a scan strategy [11, 13, 30]. The exact\nscanning pattern is characterized by the following parameters: α (precession angular radius),\nβ (spin angular radius), τprec (precession period), and τspin (spin period). The motion of the\nsatellites and the definitions of the scanning parameters are indicated in figure 1.\n\nOur simulations use maps of the celestial sphere pixelized using HEALPix2 [31] (with\nnside = 256). A sufficiently fast sampling rate is chosen so that several hits are recorded during\neach pixel crossing, so that altering this parameter does not significantly affect the results.\nWhite instrument noise of a stationary amplitude is assumed, and under this hypothesis we\nsolve the map making equation:\n\nm̂ = (ATN−1A)−1(ATN−1d). (3.1)\n\nHere the notation is such that m̂ includes the estimated maps of Stokes parameters Î , Q̂\nand Û ; A is the pointing matrix relating data samples to the map; N is the noise covariance\nmatrix in the time domain; and we denote the polarization angle of a detector ψ with respect\nto a reference axis. Individual measurements comprising the data vector d are given by\n\nSj = I(p) +Q(p) cos 2ψj + U(p) sin 2ψj + nj (3.2)\n\nwhere nj represents a stationary white noise source for observations indexed by j. Here the\nindex j (j = 1, . . . , Np) labels the observations falling into the pixel labelled by p. The\n\n2http://healpix.sourceforge.net.\n\n– 5 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nAn#-­‐Sun\t\r  direc#on\t\r  \n\nSpin\t\r  angle\t\r  β \t\n\r\n\nPrecession\t\r  \nangle\t\r  α\t\n\r\n\nτspin\t\r  \n\nτprec\t\r  \n\nFigure 1. Representation of typical satellite scanning strategy.\n\nnormalization of the noise does not matter for our purpose. The model here assumes that\nall the beams are azimuthally symmetric and identical.\n\nThe hypothesis of white instrument noise provides considerable simplification because\nin this special case the map making equation [i.e., eq. (3.1)] can be cast into a block diagonal\nform, so that the equations for different pixels decouple from each other. Each block (labelled\nby the pixel index p) takes the form\n\n\n\n\nÎ(p)\n\nQ̂(p)\n\nÛ(p)\n\n\n\n\n=\n1\n\nNp\n×\n\n\n\n\n1 〈cos 2ψj〉 〈sin 2ψj〉\n\n〈cos 2ψj〉\n1 + 〈cos 4ψj〉\n\n2\n\n〈sin 4ψj〉\n2\n\n〈sin 2ψj〉\n〈sin 4ψj〉\n\n2\n\n1− 〈cos 4ψj〉\n2\n\n\n\n\n−1\n\n×\n\n\n\n\n∑\nj Sj\n\n∑\nj Sj cos 2ψj\n\n∑\nj Sj sin 2ψj\n\n\n\n\n(3.3)\n\nwhere the hats indicate the maximum likelihood estimator, and 〈·〉 denotes the average of a\nquantity over all data samples j.\n\nComputing the maps Î(p), Q̂(p), and Û(p) as above gives the minimum variance es-\ntimator of the sky signal in the frequency band under consideration under the hypothesis\nthat the noise of each detector is white (with no correlations in time giving rise to excess\nlow-frequency noise, nor variation of the noise r.m.s. with time), that it is uncorrelated be-\ntween detectors, and that its level is identical in all detectors [32]. It also assumes no source\nof systematic errors that may require a different detector weighting to estimate each of the\nStokes parameters (and, in particular, no bandpass mismatch).\n\n– 6 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nFollowing eq. (3.2), for this map-making solution, bandpass mismatch causes the fol-\nlowing map errors\n\n\n\n\nδÎBPM\n\nδQ̂BPM\n\nδÛBPM\n\n\n\n\n=\n\n\n\n\n1 〈cos 2ψj〉 〈sin 2ψj〉\n\n〈cos 2ψj〉\n1 + 〈cos 4ψj〉\n\n2\n\n〈sin 4ψj〉\n2\n\n〈sin 2ψj〉\n〈sin 4ψj〉\n\n2\n\n1− 〈cos 4ψj〉\n2\n\n\n\n\n−1\n\n×\n\n\n\n\nδ 〈Sj〉\n\nδ 〈Sj cos 2ψj〉\n\nδ 〈Sj sin 2ψj〉\n\n\n\n\n(3.4)\n\nwhere δ 〈Sj〉 , δ 〈Sj cos 2ψj〉 , and δ 〈Sj sin 2ψj〉 are functions of the underlying sky component\nmaps. Here we assume that the normalization of the CMB component for each detector is\nperfect. This is obviously an idealization because in reality there are also systematic errors\nfrom uncorrected gain variation, but this is a separate issue that we do not analyze here.\nMoreover, since the relative gain of the detectors is calibrated using the CMB dipole, the\napproximation that the error is mostly in the relative contributions of the other components\nis a reasonable one.\n\nGiven a model of the microwave sky, the bandpass functions of the various detectors,\nand the scanning pattern on the sky, eq. (3.4) can be used to compute the bandpass mismatch\nerrors in the reconstruction of a map of Stokes parameters. For future studies of the CMB\npolarization, and in particular for the search for primordial B modes, the error of greatest\nconcern arises from the leakage of the I component of the foregrounds into the Q and U\ncomponents of the maximum likelihood band sky maps. From eq. (3.4) we observe that\nthe three terms δ 〈Sj〉 , δ 〈Sj cos 2ψj〉 , and δ 〈Sj sin 2ψj〉 can potentially induce a bias on the\npolarization Stokes parameters. The first term δ 〈Sj〉 has no impact if the maps of 〈cos 2ψ〉\nand 〈sin 2ψ〉 vanish. This is the case in particular if the detectors are arranged in sets of\nperfectly orthogonal pairs observing the sky along the same scanning path. If in addition for\neach such pair there is a matching pair observing at an angle of 45◦ relative to the first one,\nwe get an optimized configuration [32] for which the 3×3 matrix in eq. (3.3) takes the form\n\n\n\n\n1 0 0\n\n0\n1\n\n2\n0\n\n0 0\n1\n\n2\n\n\n\n\n−1\n\n. (3.5)\n\nThis simple form is preserved when observations are made with a set of such ‘optimized\nconfigurations’ oriented at any angle with respect to each other. This type of detector\narrangement was used for the Planck mission and is now standard for all proposed CMB\npolarization experiments. We then get\n\nδQ̂BPM (p) = 2δ 〈Sj cos 2ψj〉 ,\nδÛBPM (p) = 2δ 〈Sj sin 2ψj〉 , (3.6)\n\nwhere under the sky model presented in section 2\n\nδ 〈Sj cos 2ψj〉 =\n∑\n\n(c)\n\nI(c)(p)\n∑\n\ni\n\nγ(c),ifi(p) 〈cos 2ψi,j〉 ,\n\nδ 〈Sj sin 2ψj〉 =\n∑\n\n(c)\n\nI(c)(p)\n∑\n\ni\n\nγ(c),ifi(p) 〈sin 2ψi,j〉 . (3.7)\n\n– 7 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nFigure 2. Crossing moment map azimuthal averages. We show the azimuthal average of 〈cos 2ψ〉\nand 〈cos 4ψ〉 maps, constituting the totality of the component that is coherent on large angular scales.\nThe corresponding 〈sin 2ψ〉 and 〈sin 4ψ〉 maps vanish for symmetry reasons.\n\nHere the index (c) labels the non-CMB components of the sky model and i labels the detectors\nof the frequency channel under consideration (ideally supposed to have the same bandpass\nfunction). The coefficients γ(c),i vary from detector to detector as a function of the stochastic\nrealizations for the bandpass variation δνmin,i and δνmax,i. fi(p) denotes the fraction of the\ntotal hits in pixel p from the detector i, and 〈cos 2ψi,j〉 and 〈sin 2ψi,j〉 are the components of\nthe second-order crossing moments in pixel p for the detector i.\n\nBefore describing the predictions of the level of residual due to bandpass mismatch, we\nbriefly digress to examine the properties of the crossing moment maps 〈cos 2ψ〉, 〈cos 4ψ〉,\n〈sin 2ψ〉, and 〈sin 4ψ〉 for an individual detector for our model scanning pattern characterized\nby the parameter values: α = 65◦, β = 30◦, τspin = 10.002 min, and τprec = 96.2079 min.\nThose maps, which are studied into more detail in section 3.3, enter into the expression of\nthe bandpass mismatch. In ecliptic coordinates, these quantities have a nearly symmetric\npattern around the poles. Figure 2 shows the azimuthally averaged quantities (i.e., averaged\nover the ecliptic angle φ or ecliptic longitude) as a function of the sine of the latitude of\nthe maps. We observe that for a large fraction of pixels, the spin-2 and spin-4 quantities\n(functions of period π and π/2, respectively) are less then 0.2.\n\n3.1 Results\n\nWe now present numerical results for the bandpass mismatch maps and their power spectra\nbased on simulations. We construct timestreams for each detector by reading a CMB map\nand a galactic map, both at nside = 256, which were preconvolved with a symmetric Gaussian\nbeam of θFWHM = 32′. We use an instrument model with actual locations of detectors in\nthe focal plane as described in [13] or [11] depending on the case being considered. We note\nhowever that the details of the arrangement of the detectors have little or no impact on the\nleakage power spectra. The galactic map is rescaled from detector to detector using random\nerrors in the bandpass generated as described in detail in section 2. Then we construct\ncombined I, Q, and U maps obtained by applying the map making equation as given in\neq. (3.1). No noise is included in the simulation, because the map making method is linear\nand the noise does not affect the bias induced by the mismatch. For the same reason we do\nnot introduce sky emission polarization in simulations. The bandpass mismatch properties\n\n– 8 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nFigure 3. Q and U leakage maps, in ecliptic coordinates, with fiducial scanning parameters and\nNdet = 222.\n\nof each detector are generated randomly and in a statistically independent manner. Figure 3\nshows the Q and U leakage maps δQBPM and δUBPM for one particular realization. The\noutput polarization maps result from optimal map making using our simulated noiseless and\npolarizationless timestreams for the 140 GHz channel and are shown in ecliptic coordinates.\nThe simulation assumed 222 detectors, which is the number of detectors composing the\nLiteBIRD arrays described in [13], spread over a large focal plane approximatively 10 degrees\nwide observing with no HWP. The detector polarizer covers the full range of angles in the\nfocal plane with 22.5 degree separation. We assume the fiducial scanning parameters α = 65◦,\nβ = 30◦, τspin = 10 min, and τprec = 96.1803 min for the center of the focal plane (see\nsection 3.3 for a discussion of the choice of τspin and τprec to minimize the inhomogeneity of\nthe scanning pattern which is responsible for Moiré effects in the crossing moment maps).\nAt 140 GHz the bandpass mismatch error in polarization is dominated by the I component\nof the thermal dust emission, although there are subdominant contributions from the diffuse\ngalactic synchrotron emission and other non-primordial (non-CMB) components. The length\nof the survey in this simulation is exactly one sidereal year in order to ensure as uniform and\ncomplete sky coverage as possible and hence to facilitate the interpretation of those results.\nWe observe that the leakage is concentrated near the galactic plane, as expected. The bands\nat equal latitude visible in the leakage maps correspond to regions where the second order\ncrossing moments depart significantly from zero (figure 2), and as can be seen from eq. (3.7),\nthere is a strong correlation between the relative leakage amplitude and these moments.\n\nFigures 4, 5, and 6 show the bandpass mismatch leakage contributions to the EE and\nBB power spectra in different observing configurations. The power spectra are computed\nafter the 20% of the sky where the thermal dust emission is strongest is masked. The data\nin this masked region is set to zero with no apodization (which is unnecessary since the\nsmall-scale power in the leakage maps dominates over the spurious power induced by the\nmasking). For comparison we also show the CMB B mode spectrum for two different values\nof r. The dashed curves show how the signal is attenuated by beam smearing assuming the\n140 GHz FWHM beamwidth of 32’ fitted to a Gaussian profile for the present LiteBIRD\nconfiguration [33]. As will be demonstrated later, neglecting the discreteness of the scans,\nthe overall amplitude of the leakage due to bandpass mismatch is nearly Gaussian and of zero\nmean, and the variations of γdust impact all multipoles of the leakage map power spectrum in\na correlated way. For this reason, an accurate estimate of the average leakage power spectrum\n\n– 9 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nFigure 4. EE andBB leakage power spectra for α = 65◦, β = 30◦, τspin = 10 min, τprec= 96.1803 min,\nand combining data for either 74 or 222 detectors. The red curve corresponds to BB with 74 detectors,\nthe cyan to EE with 74 detectors, the blue to BB with 222 detectors and the green to EE with 222\ndetectors. The purple curve represents a model of primordial B mode power spectrum with fiducial\ncosmological parameters after Planck for r = 0.01, the black curves are including lensing for r = 0.01\nand r = 0.001. The dashed curves show the effect of convolving with a 32 arcmin beam. This plot\ndemonstrates the 1/Ndet dependance of the level of the power spectra.\n\nrequires averaging over many independent realizations even if many detectors are used for\nthe simulations. At least on large angular scales, the fluctuations in the power spectrum due\nto different realizations is roughly an overall amplitude varying as the square of a Gaussian.\n\nWe find that with all other parameters equal, the bandpass mismatch error amplitude\nscales as 1/\n\n√\nNdet where Ndet is the number of detectors (and hence the power spectrum\n\nscales as 1/Ndet). This scaling becomes more accurate when Ndet becomes large, as shown\nby comparing the EE and BB leakage power spectra for τspin = 10 min, τprec= 96.1803 min\nand Ndet of either 74 or 222. The pairs of spectra have the same shape but the ratio of power\nspectrum amplitudes is consistent with the predicted ratio 222/74 = 3.\n\nFigure 5 shows the BB power spectra for α = 65◦, β = 30◦ for several spin and\nprecession period combinations. We see that the characteristics of the leakage power spectrum\n(and in particular the location of the peaks at ` ≤ 100) depend on the exact values of τspin\nand τprec. A proper value of the ratio τprec/τspin moves the peaks in the bandpass leakage\nspectrum to higher `, away from the location of the maximum of the primordial B mode\nrecombination bump.\n\nFigure 6 compares the BB power spectra for different opening angles α and β, and also\ndifferent scan rates. With the constraint α+ β = 95◦, scan strategies with larger precession\nangle produce less leakage because they allow for more homogeneous scan angle coverage per\npixel, and hence lower |〈cos 2ψj〉| and |〈sin 2ψj〉| per individual detector.\n\n– 10 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nFigure 5. BB leakage power spectra for α = 65◦, β = 30◦, τspin=10 min, τprec=93 min (red);\nτspin=10 min, τprec=96.1803 min (green); and τspin=10/3 min, τprec=96.1803 min (blue). Simulations\ninclude 222 detectors and 365 days observation. See the figure 4 caption for a description of the model\ncurves.\n\nWe observe that the power spectra above (without a HWP) are approximately propor-\ntional to `−η where η ≈ 2.5. This angular power spectrum is less steep than that of dust\nemission itself. The shape of the resulting leakage spectrum can be expressed as a kind\nof convolution between the harmonic coefficients of the crossing moment maps and of the\ndust component map (see ref. [34] for an analytical explanation of this power law). This\nspectral shape is problematic on very large scales, for example near the reionization bump,\nbecause the ratio of the bandpass mismatch to the white noise component of the detector\nnoise (having an η ≈ 0 spectrum) increases toward lower multipole number `. We observe\nsome dependance of the amplitude of the leakage spectra with respect to the scanning strat-\negy parameters α and β. Scanning strategies with more uniform angular coverage (provided\nby larger precession angles for the studied cases) have a lower leakage amplitude.\n\nWhen the experiment observes with a rotating HWP, the equivalent of an optimized\npolarimeter configuration is straightforwardly obtained when the HWP observes a given sky\nposition p̂ during an integer number of turns (and, thus, for an evenly spread set of angles\nbetween 0 and 2π). In practice, however, the pointing direction moves while the HWP\nrotates, and hence data samples are not usually so evenly distributed. However, when the\nHWP rotates at 1.467 Hz (88 rpm) while the instrument beam scans the sky with a spin period\nof τspin = 10 minutes and with a 30◦ angle, the beam is displaced by 0.204◦ (about 12.3′)\neach time the HWP makes one turn. Neglecting this displacement, single detector timelines\nof I, Q, and U with no bandpass mismatch leakage can be straightforwardly obtained from\nthe data set, and projected onto sky maps with optimal noise averaging, i.e., equivalent\nto the generalized least square solution of eq. (3.1). Of course, a real-life HWP is not\n\n– 11 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nFigure 6. BB leakage power spectra for different scanning parameters. In cyan: α = 65◦, β = 30◦,\nτspin=10 min, τprec=96.1803 min, red: α = 50◦, β = 45◦, τspin=10 min, τprec=96.1803 min, green:\nα = 50◦, β = 45◦, τspin=2 min, τprec=4 day, blue: α = 30◦, β = 65◦, τspin=2 min, τprec=4 day.\nSpectra are computed for 222 detectors. Curves for the B mode model are described in the figure 4\ncaption. For the scanning strategies with a long precession period, we computed spectra for 100\ndetectors rescaling to 222 equivalent detectors using the 1/Ndet dependence.\n\nperfectly achromatic and hence is likely to introduce bandpass mismatch effects of its own.\nWe postpone to future work the study of this effect.\n\nTo illustrate the added value of a perfect HWP, we perform a simple set of simulations\nin which the input sky (smoothed by a 32′ beam) is a Healpix map pixelized at nside = 256.\nThe pixel size is well matched to the rotation speed of the HWP, which makes about one turn\nwhile it crosses a pixel. However, numerical effects will generate unevenness in the angular\ncoverage of each pixel, and thus, when multi-detector maps are made using eq. (3.1), small\nbandpass leakage mismatch effects will subsist. Simulating the observation of this model sky\nwith the use of a HWP spinning at 88 rpm and other parameters set to α = 65◦, β = 30◦,\nτspin =10 min, τprec = 96.1803 min, we obtain the small residual leakage shown in figure 7,\nwhich confirms the effectiveness of the HWP in reducing bandpass leakage by homogenizing\nthe angular coverage in each pixel. The shape of the spectrum of the residual is similar to\nthat of white noise. Its origin is in the small unevenness of the angle distributions across the\npixels and is an artefact of sky pixelization.\n\nWe verify that in case of a perfect HWP, the multi-detector solution for the polarization\nis close to the solution consisting in combining single detector (including the HWP) polar-\nization maps, as the residual leakage and its impact of r that can be read off the plot, is\nnegligible.\n\nTable 1 shows the contribution to r that would result from uncorrected bandpass mis-\nmatch based on its power spectrum averaged over many realizations, calculated using the\n\n– 12 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nFigure 7. EE and BB leakage power spectra with rotating HWP for α = 65◦, β = 30◦ and spin\nperiod of 10 min with a HWP rotating at 88 rpm for 50 detectors.\n\n2 ≤ ` ≤ 10 10 ≤ ` ≤ 200\n\nα = 30◦; β = 65◦; τprec = 4 days; ωspin = 0.5 rpm 1.83 ×10−3 9.32 ×10−5\n\nα = 50◦; β = 45◦; τprec = 4 days; ωspin = 0.5 rpm 6.49 ×10−4 4.66 ×10−5\n\nα = 50◦; β = 45◦; τprec = 96 min; ωspin = 0.1 rpm 6.32 ×10−4 3.08 ×10−5\n\nα = 65◦; β = 30◦; τprec = 93 min; ωspin = 0.1 rpm 3.29 ×10−4 7.61 ×10−5\n\nα = 65◦; β = 30◦; τprec = 96 min; ωspin = 0.1 rpm 3.27 ×10−4 2.11 ×10−5\n\nα = 65◦; β = 30◦; τprec = 96 min; ωspin = 0.3 rpm 3.03 ×10−4 1.77 ×10−5\n\nTable 1. Contribution of bandpass mismatch error to the tensor-to-scalar ratio r computed according\nto eq. (3.8). The level of the bandpass leakage relative to primordial B mode signals is acceptable at\nthe angular scale of the recombination bump, but problematic for the reionization bump at ` . 10.\nScanning strategies with larger α and smaller β perform better, as they provide more uniform angular\ncoverage in each pixel.\n\nprojection\n\nδ̂r =\n\n∑`max\n`=`min\n\n(2`+ 1)C`Ĉ`∑`max\n`=`min\n\n(2`+ 1)C2\n`\n\n. (3.8)\n\nHere C` is the power spectrum for the primordial B mode signal normalized to r = 1. The\ntable shows δr calculated for two ranges of `: one with ` ∈ [2, 10] to isolate the signal from\nthe re-ionization bump, and another with ` ∈ [10, 100] to isolate the signal arising from the\nrecombination bump. The results in the table assume Ndet = 222 detectors, but can be\nrescaled based on the 1/Ndet dependence to other numbers of detectors. These results are\nonly an order of magnitude estimate because they are based on a single 140 GHz channel,\n\n– 13 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nand it has been assumed that very low and very high frequency channels have been used to\nremoved the non-primordial components completely. We stress that the bandpass mismatch\npower spectrum is not a simple bias that can be predicted and subtracted away because its\noverall amplitude suffers large fluctuations, which are of the same order of magnitude as the\naverage bias itself.\n\n3.2 Analytic estimates\n\nWith the objective of finding fast and easy ways to predict the magnitude of potential leakage\nwithout running many Monte Carlo simulations, and in order to understand how the patterns\nshown in the leakage map in figure 3 are related to the scanning strategy, we now study\ntheoretically in more detail how the leakage manifests itself in the polarization maps. To this\nend, we expand the solution of the map making equation [eq. (3.1)].\n\nWe derive a simple expression for the leakage originating from differencing the signal\nfrom a pair of orthogonally polarized detectors observing instantaneously at the same location\nin the sky, so that data of the two detectors of the pair i at time t in pixel p denoted as\nSi;a(t) and Si;b(t) are given by\n\nSi;a(t) = Ii;p +Qp cos 2ψ(t) + Up sin 2ψ(t) +Mi;p,\nSi;b(t) = Ii;p −Qp cos 2ψ(t)− Up sin 2ψ(t)−Mi;p. (3.9)\n\nHere we assume no noise and perfect calibration on the CMB (e.g., using the CMB dipole),\nand ψ is the polarizer angle for detector a. Ii;p, Qp, Up are the Stokes parameters of the sky\nsignal, Ii;p being the mean intensity parameter for the detector pair i, and Mi;p represents\nthe bandpass mismatch component, which is given by\n\nMi;p =\n1\n\n2\n\n∑\n\n(c)\n\n(\nγa(c) − γb(c)\n\n)\nIp,(c). (3.10)\n\nHere the index (c) labels the non-CMB sky components. The coefficient differences\n(\nγa(c) −\n\nγb(c)\n)\n\nvary from detector pair to detector pair, as explained in section 2 (see in particular\n\neq. (2.3)). To minimize clutter, we have suppressed the index i labelling the detector pairs.\nWe neglect the subdominant effect of bandpass mismatch on the polarized sky components.\nAs in the previous section, we neglect noise in our analysis. The estimated noiseless Stokes\nparameter maps Q̂p and Ûp can be expanded as Q̂p = Qp + δQp and Ûp = Up + δUp, where\nδQ and δU represent the leakages to polarization resulting from bandpass mismatch. Ideal\nsolutions with no leakage are given in eq. (3.9).\n\nThe map making equation gives\n\n\n\n\nÎp\n\nQ̂p\n\nÛp\n\n\n =\n\n\n\n\n1 0 0\n\n0 1\n2 (1 + 〈cos 4ψ〉) 1\n\n2〈sin 4ψ〉\n0 1\n\n2〈sin 4ψ〉 1\n2 (1− 〈cos 4ψ〉)\n\n\n\n\n−1\n\n\n〈S〉\n〈12(Sa − Sb) cos 2ψ〉\n〈12(Sa − Sb) sin 2ψ〉\n\n\n , (3.11)\n\nand the zeros in the 3×3 matrix result because the exact orthogonality of the two detectors\nof each pair insures that 〈cos 2ψ〉 and 〈sin 2ψ〉 vanish exactly [compare with eq. (3.4)], so\nthat the expression for Îp decouples from the expressions for Q̂p and Ûp. Consequently, the\n\n– 14 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nFigure 8. Left: leakage for the Q component relative to the dust temperature (δQ/IGal) after\npolarization reconstruction using one bolometer pair only and a one year observation time. Right:\naveraged cos 2ψ in each pixel for one bolometer after one year observation time. This quantity is\nstrongly correlated with the relative leakage Q component with respect to the dust intensity.\n\nleakages are given by\n\n(\nδQp\nδUp\n\n)\n=\n\n(\n1\n2(1 + 〈cos 4ψ)〉 1\n\n2〈sin 4ψ〉\n1\n2〈sin 4ψ〉 1\n\n2(1− 〈cos 4ψ)〉\n\n)−1(〈Mp cos 2ψ〉\n〈Mp sin 2ψ〉\n\n)\n\n=\n2\n\n(1− 〈cos 4ψ〉2 − 〈sin 4ψ〉2)\n\n(\n1 + 〈cos 4ψ〉 −〈sin 4ψ〉\n−〈sin 4ψ〉 1− 〈cos 4ψ〉\n\n)(\n〈Mp cos 2ψ〉\n〈Mp sin 2ψ〉\n\n)\n. (3.12)\n\nAssuming that 〈cos 4ψ〉2 + 〈sin 4ψ〉2 � 1 (which is not so bad an approximation except very\nnear the poles), we obtain\n\n(\nδQp\nδUp\n\n)\n≈ 2\n\n(\n〈Mp cos 2ψ〉\n〈Mp sin 2ψ〉\n\n)\n. (3.13)\n\nFor one Galactic component, by replacing Mp by its expression in eq. (3.10), the relative\namplitude of the leakage can be written as\n\n(\nδQp\n\nIGal;p\nδUp\n\nIGal;p\n\n)\n=\n(\nγaGal − γbGal\n\n)(〈cos 2ψ〉\n〈sin 2ψ〉\n\n)\n. (3.14)\n\nThe term on the right-hand side is one of the crossing moment terms for a single detector.\nWe should then observe a large correlation between the two maps on the two sides of the\nequation. We have verified, with the help of simulations of data for one detector pair, this\nrelationship for two different scanning strategies: α = 65◦ and β = 30◦ and α = 50◦ and\nβ = 45◦. Figure 8 shows the relative leakage map δQp/IGal;p and the quantity\n\n∑\ncos 2ψ/np.\n\nThe U component (not shown here) exhibits similar properties.\nFigure 9 shows the correlation of the two maps by plotting the values of one map versus\n\nthe other for a subset of pixels. We observe a high correlation between the two maps. We\nverify that the slope is given by the coefficient ∆γ = γa − γb as derived in eq. (3.14). This\nfigure shows the tight link between the crossing moments and the relative leakage due to\nbandpass mismatch. It also shows that the approximations made to derive eq. (3.14) are\nvalid since we observe a relatively small scatter around the linear slope. The outliers in the\nfigure are due to pixels near the ecliptic poles where the angle coverage is less uniform for\nthe scanning parameters used as a baseline in this work.\n\n– 15 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nFigure 9. Values of the relative leakage δQp/IGal;p for a pair of detectors with orthogonal polariza-\ntions of a function of the scanning strategy parameter (1/np)\n\n∑\ncos 2ψ (see text) after map making\n\nwith two detectors only. We observe a tight correlation between the relative leakage and the second\norder crossing moments.\n\nWe now consider the solution combining more detectors. The generalization of eq. (3.13)\ngives for the resulting leakage component\n\n(\nδQp\nδUp\n\n)\n=\n\n\n\n\n1\n2\n\n∑\ni\n\n∑\nj\n\n(1 + cos 4ψji )\n1\n2\n\n∑\ni\n\n∑\nj\n\nsin 4ψji\n\n1\n2\n\n∑\ni\n\n∑\nj\n\nsin 4ψji\n1\n2\n\n∑\ni\n\n∑\nj\n\n(1− cos 4ψji )\n\n\n\n\n−1\n\n\n∑\ni\n\n∑\nj\n\ncos 2ψji Mi,p\n\n∑\ni\n\n∑\nj\n\nsin 2ψji Mi,p\n\n\n (3.15)\n\nwhere we sum over all the detector pairs indexed by i and over all samples j falling in pixel p\nfor each detector. In this case, for which we consider the realistic configuration of more than\none pair of detectors per pixel, the covariance matrix above becomes nearly diagonal. As\nthe number of detectors is increased, the matrix in eq. (3.15) becomes increasingly diagonal.\nThe total leakage is then simply, replacing the leakage term Mp by its expression:\n\nδQp\nIGal;p\n\n≈ 2\n\nNhit\n\n∑\n\ni\n\n∆γi\n∑\n\nj\n\ncos 2ψji , (3.16)\n\nusing eq. (3.10), where we have defined Nhit as the total number of hits including all detectors\n(and not only count 1 per detector pair which explains the cancellation of the 1/2 factors\nsince the sum runs over detector pairs), and ∆γi = γai − γbi . The leakage vanishes if each\nindividual detector has uniform angle coverage. We observe that the relevant quantities to\nestimate the level of leakage for a given scanning strategy are the individual detector second\norder crossing moments. Following our hypothesis that the γ parameters are random and\n\n– 16 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nFigure 10. Left: estimated leakage variance of the Q component relative to the dust temperature\n(Var (δQp/IGal;p)) after polarization reconstruction using all bolometer pairs and one year of observa-\ntions. We used 10 independent realizations of the bandpass to estimate the variance. Right: averaged〈(\n\n(1/n̄p)\n∑\n\ncos 2ψj\ni\n\n)2〉\ndet\n\nin each pixel for all bolometers after one year observation time. As for the\n\ndetector pair case, we observe a tight correlation of the two maps on large angular scales.\n\nuncorrelated, we express the variance of the leakage map as:\n\nVar\n\n(\nδQp\nIGal;p\n\n)\n≈\n∑\n\ni\n\nVar(∆γi)\n\n\n∑\n\nj\n\ncos 2ψji\n\n\n\n\n2(\n2\n\nNhit\n\n)2\n\n, (3.17)\n\nwhich gives, since Var(∆γ) = 2Var(γ):\n\nVar\n\n(\nδQp\nIGal;p\n\n)\n≈ 4\n\nVar(γ)\n\nNdet\n\n〈(∑\ncos 2ψji\nn̄p\n\n)2〉\n\ndet\n\n, (3.18)\n\nwhere 〈 · 〉det denotes average over all detectors, and n̄p = Nhit\nNdet\n\nis the average number of\nhits per detector. The expression for the U component is similar with the cosine replaced by\na sine. This expression for the variance of the leakage map is also valid if detectors are not\narranged by pairs.\n\nFigure 10 compares the maps of the variance on the left-hand side of the previous rela-\ntionship which was estimated with ten independent realizations of the bandpass parameters,\n\nwith the quantity\n〈(\n\n(1/n̄p)\n∑\n\ncos 2ψi\n\n)2〉\ndet\n\n. Figure 11 shows the correlations between the\n\ntwo quantities on a scatter plot. We observe a significant correlation of the two quantities,\nespecially on large scales. The dispersion is partly due to the limited number of realizations\nto estimate the variance. Nevertheless, this shows that the level of leakage can be evalu-\nated by estimating the second order crossing moments only for different scanning strategies\nwithout the need of running large simulations. This result explains what was observed in\nfigure 6, showing the level of the leakage with respect to the scanning parameters α and β.\nThe strategies with more uniform angle distribution (the ones with larger precession angle)\nshow lower residuals (see also [35] for the link with other systematic effects).\n\nResults show that contamination from bandpass mismatch even if small could contribute\nto the B mode spectrum at a non-negligible level, close to the detection limit of primordial B\nmodes with future satellite missions. Systematic variation of the bandpass functions across\nthe focal plane, as opposed to the uncorrelated random variations assumed in this study,\ncould produce larger errors. These considerations motivate developing correction methods,\nwhich we present in the companion paper [20].\n\n– 17 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nFigure 11. Estimated variance distribution of the relative leakage parameter: Var (δQp/IGal;p) as a\n\nfunction of c =\n〈(\n\n(1/n̄p)\n∑\n\ncos 2ψt;i\n\n)2〉\ndet\n\n(see text) after map making including all detectors. We\n\nhave averaged over ten realizations to estimate the variance.\n\n3.3 Importance of avoiding resonances\n\nHere we briefly explain some considerations for choosing the scan frequency parameters ωspin\n\nand ωprec. We found that to obtain good crossing moment maps, careful attention must\nbe paid to choosing the ratios of the hierarchy of scan frequencies ωann � ωprec � ωspin,\nand when there is a continuously rotating HWP also ωHWP. For ωprec/ωann, we choose to\nmake this number an integer so that the scan pattern closes. In all the simulations reported\nhere, we assumed a single survey of exactly one year in duration. Given the large number\nof precession cycles in a year, this requirement can be achieved by means of a very small\nadjustment in ωprec. One might also want to do the same for the spin period, but this is less\ncritical because of its shortness compared to a year.\n\nMore critical is the ratio θ = ωspin/ωprec, which must be chosen so that θ cannot be well\napproximated by simple fractions of the form p/q where p and q are relatively prime and q is\nsmall in a sense that we shall make more precise shortly. Of concern are exact or near exact\nresonances where q is less than of order ωprec/ωspin.\n\nBefore entering into the theory of how the ratio θ should be chosen (and jumping ahead\nslightly), we show what goes wrong when θ is not well chosen. For example, our first try\nhad τspin = 10 min and τprec = 93 min and gave hit count and crossing moment maps with\nclearly visible Moiré patterns at intermediate angular scales, as shown in figure 12, showing\nclear evidence of a near resonance. However, when ωprec was sped up by the Golden ratio\nΦ = (1 +\n\n√\n5)/2 = 1.61803398875 (reputed to be the “most irrational” number),3 these\n\n3See for example Michael Berry, (1978, September), Regular and irregular motion, in S. Jorna (Ed.), AIP\nConference proceedings (Vol. 46, No. 1, 16–120), AIP for a nice discussion of these questions in a different\ncontext, that of perturbations of integrable systems in classical mechanics, KAM theory, and the stability of\nthe solar system.\n\n– 18 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nFigure 12. Effect of a poorly chosen scanning frequency ratios. The map on the left has θ =\nωspin/ωprec = 9.3, whose continued fraction representation is [9, 3, 3], whereas the lower map has the\nmore irrational ratio θ = 9.61803, whose continued fraction representation is [9, 1, 1, 1, ......]. A series\nof Moiré patterns on intermediate angular scales is clearly visible in the map on the left, which lead\nto spikes in the crossing moment map power spectra, and also in the final bandpass mismatch power\nspectra. The artefacts can be avoided by choosing ratios of frequencies judiciously in order to avoid\ngood rational approximations.\n\nundesirable Moiré patterns disappear, as shown in the bottom right panel of the figure. The\nsame effect could be achieved by altering the ratio θ by just 5%, so that the spin cycle has the\nsame phase as with the Golden ratio sped up. We note that the effect of these Moiré patterns\non the bandpass mismatch power spectra is to introduce peaks at multipole numbers at which\nthe bandpass mismatch error is increased by up to about an order of magnitude beyond the\nbaseline, where it would be if θ had been well chosen to avoid near resonances. We also note\nthat when a continuously rotating HWP is introduced, there are two independent ratios to\nworry about, although the artefacts are less acute than in the case of no rotating HWP.\n\nThe theory of choosing ratios to avoid near resonances relates to problems well studied\nby pure mathematicians in the area of number theory, or more specifically the theory of\nDiophantine approximations, and we discussed these issues in more detail elsewhere [36].\nThe tool for characterizing the near resonance properties of real numbers is the continued\nfraction representation, where we expand\n\nθ = [a0, a1, a2, . . .] = ao +\n1\n\na1 +\n1\n\na2 + . . .\n\n(3.19)\n\nwhere a0 is an integer and a1, a2, . . . are positive integers. For a rational number, the contin-\nued fraction representation terminates; for an irrational number it is of infinite length. The\npartial sums, known as ‘convergents,’ generate a sequence of ‘best rational approximations’\np/q to θ,4 with q ascending. When a coefficient an is large compared to one, the preceding\nconvergent is a particularly good approximation to θ considering the magnitude of q. The\nGolden ratio Φ has the continued fraction representation [1, 1, 1, . . .], and thus has among the\nworst approximation properties of any number.\n\nFor the parameters used in the simulations reported below, we adjusted the precession\nperiod so that there are an integer number 5467 cycles in a sidereal year, giving a precession\n\n4An irreducible fraction p/q is a ‘best approximation’ to θ if |θ − p′/q′| > |θ − p/q| whenever q′ < q.\n\n– 19 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nperiod of 96.2080 minutes, and we replaced the ratio of θ = 9.3, which in terms of contin-\nued fractions is represented by [9, 3, 3], with the ratio 9.618033988749895,5 whose continued\nfraction representation is [9, 1, 1, 1, ......], giving a spin period of 10.002876 minutes. One may\nask: approximately to what accuracy would one wish to maintain this ratio? Certainly more\naccuracy than the inverse of the number of precession cycles in a year would be superflu-\nous. In fact, less accuracy would be adequate, the exact number depending on the precise\nscanning parameters, but we postpone further assessment of the required precision to future\nwork. Moreover, it is more the absolute pointing that matters and not so much a question\nof maintaining precise ratios at any particular moment.\n\nAn important practical question is what accuracy is required in the ratios of the fre-\nquencies in order to avoid the Moiré patterns due to near resonances. It is not possible to\nprovide a general answer to this question, but we performed some numerical experiments for\nthe scanning frequencies considered in this paper and found that tuning the ratio of the fre-\nquencies to about 0.2% sufficed. It should be stressed that it is the relative phase rather than\nthe instantaneous ratio of frequencies that matters for avoiding Moiré artifacts. In the above\ndiscussion we considered only a single ratio, but for more complicated situations with several\nfrequencies, there is more than one ratio to keep away from near resonant values. A rotating\nhalf-wave plate, for example, introduces another frequency, and in principle the annual drift\nalso allows other dimensionless ratios of frequencies to be formed. These complications will\nbe investigated elsewhere.\n\n3.4 Hitcount and crossing moment map properties\n\nWe now examine the properties of the hitcount map Ha(p) for a particular detector labeled\nby the index a (where the index p denotes a particular discrete pixel) as well as maps of\n〈cos 2ψ(p)〉a , 〈sin 2ψ(p)〉a , 〈cos 4ψ(p)〉a , and 〈sin 4ψ(p)〉a , which, as already stressed enter\ninto the expressions for the bandpass mismatch.\n\nFigures 13 and 14 show the maps Ha(p), 〈cos 2ψ(p)〉a , 〈sin 2ψ(p)〉a , 〈cos 4ψ(p)〉a , and\n〈sin 4ψ(p)〉a for a typical detector with the fiducial scan parameters given above for a full-year\nscan (so that there are no boundaries).\n\nThese figures demonstrate that in all the maps (except for the 〈sin 2ψ(p)〉a , and\n〈sin 4ψ(p)〉a maps), when small-scale structure is ignored, there is an azimuthally symmetric\nnon-uniformity. From the azimuthally averaged quantities shown in figure 2, we can see that\nsuperimposed on this azimuthally symmetric component is a component almost completely\ndevoid of large-angle power resulting from the discreteness of the scans. Figure 15 shows\nthe power spectra of the crossing moment maps. We note that given the finite size of the\nfocal plane, the spin opening angle β varies from detector to detector. This variation in β\ninduces an azimuthally symmetric component having large-scale power in the difference map\nof moments for different detectors at different locations in the focal plane. Also present will\nbe a small-scale component, which would disappear in the limit ωspin, ωprec → +∞ along\nwith the sampling rate while keeping the ratio ωspin/ωprec fixed. This small scale power is\nsomewhat akin to shot noise.\n\n5In any specific application, the objective of avoiding near resonances obviously requires an accuracy involv-\ning only a finite number of terms of the continued fraction expansion. Moreover, it is less the instantaneous\nratio of frequencies that matters but rather the relative phase. We have found using numerical simulations\nthat avoiding Moiré patterns is achieved when the ratios are maintained with a relative accuracy of 1 part in\n103, although the exact accuracy needed will depend on the particular application.\n\n– 20 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nFigure 13. Hitcount map and azimuthal average for fiducial scanning pattern. The hitcount map is\nroughly uniform except for some localized spikes of high density around the ecliptic poles and at the\ncaustics at ecliptic latitude ±(α− β) = ±(65◦ − 30◦) = ±35◦. In the bottom plot the horizontal axis\nis cos θ where θ is the angle from the north ecliptic pole.\n\nFigure 14. Crossing moment maps for the fiducial scanning pattern. The four relevant crossing\nmoment maps 〈cos 2ψ〉 , 〈cos 4ψ〉 , 〈sin 2ψ〉 , 〈sin 4ψ〉 (left to right, top to bottom) are shown for the\nfiducial scanning pattern (defined in the text) for a single detector whose polarization axis is oriented\nalong the line running from the center of the beam to the spin axis. The corresponding maps for other\npolarizer orientations can be obtained trivially using the property that the first two maps transform as\na spin-2 vector and the second two as a spin-4 vector under rotations of the polarization orientation.\nWe observe that the cosine maps have structures coherent on large scales and azimuthally symmetric\nin ecliptic coordinates, whereas the sine maps include only small-scale noise (which is also present in\nthe cosine maps) but have no structure coherent on large angular scales.\n\n– 21 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nFigure 15. Crossing moment map power spectra. We show the power spectra of the maps of figure 14.\nThe spectra of the two cosine maps, because of the azimuthally symmetric large power coherent on\nlarge scales shown in figure 2, have power spectra scaling similar to `−2 for the even moments, whereas\nthe two sine maps (bottom) row exhibit spectra resembling pure white noise.\n\nWe now consider the effect of a continuously rotating HWP on the second- and fourth-\norder crossing moment maps, as shown in figures 16 and 17. We see that the azimuthally\nsymmetric structures coherent on large angular scales disappear as a consequence of the\ncontinuously rotating HWP. The main consequence is to beat down by many orders of mag-\nnitude the (`−2.5)-like power present on large angular scales in cosine maps, but there is also\nsubstantial reduction in the power at all multipole numbers compared to the no-HWP case.\n\nWe point out that much the same beneficial effect could also be obtained using a dis-\ncretely stepped HWP (with a stepping pattern tailored to produce the necessary cancel-\nlations). Alternatively, less complete cancellations could also be obtained by stepping the\norientation of the focal plane about its optical axis. These rotations are called “deck rota-\ntions” in the BICEP2 papers (see e.g., [8]), a terminology that we shall also adopt. Allowing\nfor such deck rotations, however, would also require additional complexity in the satellite\ndesign beyond the simplest no HWP design. Moreover, for the deck rotations alone, the\ncancellations would be imperfect because the values of β for the individual detector scanning\npatterns change as the focal plane is rotated (except possibly for one detector situated at the\noptical axis, assumed to coincide with the deck rotation axis).\n\n– 22 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nFigure 16. Crossing moment maps (with rotating HWP). We show the same maps as in figure 14\nexcept that there is a rotating HWP, as explained in the main text. We observe that the coherent\npower in the cosine maps has been scrambled as a result of the presence of the HWP and the overall\npower in all the maps has greatly been reduced.\n\nFigure 17. Crossing moment map power spectra (with rotating HWP). We show the power spectra\nfor the maps in figure 16. The power spectra of the 〈cos 2ψ〉 and 〈cos 4ψ〉 have a white noise-like\nspectrum rather than an (`−2)-like spectrum because the HWP has scrambled azimuthally symmetric\ncomponent coherent on large-scale present in the case with no HWP.\n\n– 23 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\n4 Conclusions\n\nThis paper presented estimates of the contribution of bandpass mismatch error to the final\ndetermination of the tensor-to-scalar ratio r, both for the window situated at the ‘re-ionization\nbump,’ and for the window at the ‘recombination bump’, for a set of observation strategies\nconsidered for future CMB polarization experiments. In the case without a HWP, requiring in\nthe optimal case the combination of multi-detector data, we show that the bandpass mismatch\nerror in polarization has a red power spectrum resembling `−2.5. The contribution to r is of\nthe order of 10−3 at the reionization bump, assuming random variations of the detector filters\nfor typical arrays at 140 GHz, such that the variation of the dust component amplitude is of\nthe order of 0.6 %. However, with a continuously rotating HWP the spectrum is similar to\nthat of white noise, with the power on the largest scales many orders of magnitude smaller\nthan without a HWP. This is due to the fact that an ideal HWP allows nearly uniform\nangle coverage in each pixel, and hence the multi-detector solution is almost equivalent to\nthe combination of single detector maps of Q and U . The HWP also cancels correlations\nin the non-uniformity in the angular coverage between different pixels, hence the efficient\nreduction in power of the bandpass mismatch on large angular scales. We further note\nthat a stepped HWP would reduce bandpass mismatch in a similar way provided that its\ndiscrete rotations are properly synchronized with the scan pattern. We show that even with\na simplistic multi-detector map-making approach, the HWP suppresses the bandpass leakage\npower by several orders of magnitude on large scales. We note however that this conclusion\nignores the problem of HWP imperfections, in particular chromaticity effects, which would\ngenerate bandpass mismatch systematics of its own.\n\nTo obtain accurate estimates of the bandpass mismatch error, more precise information\nwould be needed concerning (1) the scan pattern assumed, (2) the variations in the bandpass\nfunctions from detector to detector, and (3) the foreground removal process. For (1) we\nused one of the LiteBIRD candidate scan patterns. Likewise, for (2) we based our model for\nvariations in the bandpass function from preliminary results that have actually been achieved\nin the laboratory between different detectors without a HWP, but there may be effects not\nproperly taken into account that could lead to larger errors, or conversely further technological\ndevelopment could lead to reduced mismatch between bandpass functions. With respect to\n(3), we simply calculated the bandpass error in a 140 GHz map, assuming that but for this\nerror, the dominant dust and synchrotron components could be removed by subtraction using\na perfect foreground component templates. This is certainly a simplification which provides\na simple estimate that can be described in a simple term. If the foregrounds turn out to\nbe very complicated, the CMB clean map might be the result of a linear combination of\nmaps whose coefficients (or varying sign) are much larger than one (assuming the maps are\nnormalized to the CMB). A foreground cleaning of this sort (if necessary) may lead to larger\nbandpass errors than our estimate. Finally, we mention one caveat of our analysis: we did\nnot include 1/f noise in our modeling, a feature that allowed us to carry out pixel-by-pixel\nmap making and avoid including extra model parameters.\n\nIn this paper we have estimated bandpass mismatch error assuming that no measures\nhave been taken to correct for or otherwise mitigate this systematic error. In the companion\npaper ref. [20] we explore paths to correct for and mitigate bandpass mismatch error with a\ndedicated data processing step.\n\n– 24 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\nAcknowledgments\n\nDuc Thuong Hoang thanks the Vietnam International Education Cooperation Department\n(VIED) of the Ministry of Education and Training for support through a Ph.D. fellowship\ngrant. We thank Aritoki Suzuki for useful discussions and sharing with us data on filter\nbandpass measurements.\n\nReferences\n\n[1] Planck collaboration, P.A.R. Ade et al., Planck Early Results. I. The Planck mission, Astron.\nAstrophys. 536 (2011) A1 [arXiv:1101.2022] [INSPIRE].\n\n[2] WMAP collaboration, C.L. Bennett et al., Nine-year Wilkinson Microwave Anisotropy Probe\n(WMAP) Observations: Final Maps and Results, Astrophys. J. Suppl. 208 (2013) 20\n[arXiv:1212.5225] [INSPIRE].\n\n[3] Planck collaboration, N. Aghanim et al., Planck 2015 results. XI. CMB power spectra,\nlikelihoods and robustness of parameters, Astron. Astrophys. 594 (2016) A11\n[arXiv:1507.02704] [INSPIRE].\n\n[4] ACTPol collaboration, T. Louis et al., The Atacama Cosmology Telescope: Two-Season\nACTPol Spectra and Parameters, JCAP 06 (2017) 031 [arXiv:1610.02360] [INSPIRE].\n\n[5] SPT collaboration, A.T. Crites et al., Measurements of E-Mode Polarization and\nTemperature-E-Mode Correlation in the Cosmic Microwave Background from 100 Square\nDegrees of SPTpol Data, Astrophys. J. 805 (2015) 36 [arXiv:1411.1042] [INSPIRE].\n\n[6] POLARBEAR collaboration, P.A.R. Ade et al., A Measurement of the Cosmic Microwave\nBackground B-Mode Polarization Power Spectrum at Sub-Degree Scales with POLARBEAR,\nAstrophys. J. 794 (2014) 171 [arXiv:1403.2369] [INSPIRE].\n\n[7] SPT collaboration, R. Keisler et al., Measurements of Sub-degree B-mode Polarization in the\nCosmic Microwave Background from 100 Square Degrees of SPTpol Data, Astrophys. J. 807\n(2015) 151 [arXiv:1503.02315] [INSPIRE].\n\n[8] BICEP2, Keck Array collaboration, P.A.R. Ade et al., BICEP2 /Keck Array V:\nMeasurements of B-mode Polarization at Degree Angular Scales and 150 GHz by the Keck\nArray, Astrophys. J. 811 (2015) 126 [arXiv:1502.00643] [INSPIRE].\n\n[9] BICEP2, Planck collaboration, P.A.R. Ade et al., Joint Analysis of BICEP2/Keck Array\nand Planck Data, Phys. Rev. Lett. 114 (2015) 101301 [arXiv:1502.00612] [INSPIRE].\n\n[10] BICEP2, Keck Array collaboration, P.A.R. Ade et al., Improved Constraints on Cosmology\nand Foregrounds from BICEP2 and Keck Array Cosmic Microwave Background Data with\nInclusion of 95GHz Band, Phys. Rev. Lett. 116 (2016) 031302 [arXiv:1510.09217] [INSPIRE].\n\n[11] CORE collaboration, J. Delabrouille et al., Exploring Cosmic Origins with CORE: Survey\nrequirements and mission design, arXiv:1706.04516 [INSPIRE].\n\n[12] M. Hazumi et al., LiteBIRD: a small satellite for the study of B-mode polarization and\ninflation from cosmic background radiation detection, Proc. SPIE Int. Soc. Opt. Eng. 8442\n(2012) 844219 [INSPIRE].\n\n[13] T. Matsumura et al., LiteBIRD: Mission Overview and Focal Plane Layout, J. Low. Temp.\nPhys. 184 (2016) 824 [INSPIRE].\n\n[14] A. Kogut, J. Chluba, D. J. Fixsen, S. Meyer and D. Spergel, The primordial inflation explorer\n(PIXIE), Proc. SPIE 9904 (2016) 99040W.\n\n[15] Planck collaboration, N. Aghanim et al., Planck intermediate results. XLVI. Reduction of\nlarge-scale systematic effects in HFI polarization maps and estimation of the reionization\noptical depth, Astron. Astrophys. 596 (2016) A107 [arXiv:1605.02985] [INSPIRE].\n\n– 25 –\n\n\n\nJ\nC\nA\nP\n1\n2\n(\n2\n0\n1\n7\n)\n0\n1\n5\n\n[16] WMAP Science Team collaboration, E. Komatsu et al., Results from the Wilkinson\nMicrowave Anisotropy Probe, PTEP 2014 (2014) 06B102 [arXiv:1404.5415] [INSPIRE].\n\n[17] the ABS collaboration, T. Essinger-Hileman et al., Systematic effects from an\nambient-temperature, continuously rotating half-wave plate, Rev. Sci. Instrum. 87 (2016)\n094503 [arXiv:1601.05901] [INSPIRE].\n\n[18] T. Matsumura, A cosmic microwave background radiation polarimeter using superconducting\nmagnetic bearings, Ph.D. Thesis, University of Minnesota, Minnesota U.S.A. (2006).\n\n[19] CORE collaboration, P. Natoli et al., Exploring cosmic origins with CORE: mitigation of\nsystematic effects, arXiv:1707.04224 [INSPIRE].\n\n[20] R. Banerji et al., Bandpass mismatch error for satellite CMB experiments II: Correcting the\nspurious signal, in preparation.\n\n[21] Planck collaboration, P.A.R. Ade et al., Planck 2013 results. XIII. Galactic CO emission,\nAstron. Astrophys. 571 (2014) A13 [arXiv:1303.5073] [INSPIRE].\n\n[22] Planck collaboration, A. Abergel et al., Planck 2013 results. XI. All-sky model of thermal\ndust emission, Astron. Astrophys. 571 (2014) A11 [arXiv:1312.1300] [INSPIRE].\n\n[23] Planck collaboration, P.A.R. Ade et al., Planck intermediate results. XXII. Frequency\ndependence of thermalemissionfromGalacticdustinintensity and polarization, Astron. Astrophys.\n576 (2015) A107 [arXiv:1405.0874] [INSPIRE].\n\n[24] Planck collaboration, R. Adam et al., Planck 2015 results. X. Diffuse component separation:\nForeground maps, Astron. Astrophys. 594 (2016) A10 [arXiv:1502.01588] [INSPIRE].\n\n[25] Planck collaboration, P.A.R. Ade et al., Planck 2013 results. IX. HFI spectral response,\nAstron. Astrophys. 571 (2014) A9 [arXiv:1303.5070] [INSPIRE].\n\n[26] Planck collaboration, R. Adam et al., Planck 2015 results. VIII. High Frequency Instrument\ndata processing: Calibration and maps, Astron. Astrophys. 594 (2016) A8 [arXiv:1502.01587]\n[INSPIRE].\n\n[27] WMAP collaboration, N. Jarosik et al., Three-year Wilkinson Microwave Anisotropy Pr\n(WMAP) observations: beam profiles, data processing, radiometer characterization and\nsystematic error limits, Astrophys. J. Suppl. 170 (2007) 263 [astro-ph/0603452] [INSPIRE].\n\n[28] B. Westbrook et al., Development of the Next Generation of Multi-chroic Antenna-Coupled\nTransition Edge Sensor Detectors for CMB Polarimetry, J. Low. Temp. Phys. 184 (2016) 74.\n\n[29] Planck collaboration, N. Aghanim et al., Planck intermediate results. XLVI. Reduction of\nlarge-scale systematic effects in HFI polarization maps and estimation of the reionization\noptical depth, Astron. Astrophys. 596 (2016) A107 [arXiv:1605.02985] [INSPIRE].\n\n[30] EPIC collaboration, J. Bock et al., Study of the Experimental Probe of Inflationary Cosmology\n(EPIC)-Intemediate Mission for NASA’s Einstein Inflation Probe, arXiv:0906.1188\n[INSPIRE].\n\n[31] K.M. Górski et al., HEALPix: A Framework for high resolution discretization and fast analysis\nof data distributed on the sphere, Astrophys. J. 622 (2005) 759 [astro-ph/0409513] [INSPIRE].\n\n[32] F. Couchot, J. Delabrouille, J. Kaplan and B. Revenu, Optimized polarimeter configurations for\nmeasuring the Stokes parameters of the cosmic microwave background radiation, Astron.\nAstrophys. Suppl. Ser. 135 (1999) 579 [astro-ph/9807080] [INSPIRE].\n\n[33] H. Ishino et al., LiteBIRD: lite satellite for the study of B-mode polarization and inflation from\ncosmic microwave background radiation detection, Proc. SPIE 9904 (2016) 99040X.\n\n[34] M. Bucher et al., Predicting bandpass mismatch power spectra, in preparation.\n\n[35] C.G.R. Wallis, M.L. Brown, R.A. Battye and J. Delabrouille, Optimal scan strategies for future\nCMB satellite experiments, Mon. Not. Roy. Astron. Soc. 466 (2017) 425 [arXiv:1604.02290]\n[INSPIRE].\n\n[36] M. Bucher, Maximum irrationality for optimal scanning of the CMB Sky, in preparation.\n\n– 26 –\n\n\n\nEvaluating the level of the bandpass mismatch systematic effect for the future\nCMB satellites\n\nDUC THUONG HOANG 1,2\n\n1Laboratoire Astroparticule et Cosmologie (APC), Université Paris Diderot, CNRS/IN2P3, CEA/Irfu,\nObservatoire de Paris, Sorbonne Paris Cité, 10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex\n\n13, France.\n2Department of Space and Aeronautics, University of Science and Technology of Hanoi (USTH),\n\nVietnam Academy of Science and Technology (VAST), 18 Hoang Quoc Viet, Cau Giay, Hanoi, VietNam\n\nBandpass mismatch error is one of the important systematic effects that can affect current and\nnext generation measurement of the polarization of the Cosmic Microwave Background radi-\nation (CMB). The slightly different frequency bandpasses among detectors introduce leakage\nfrom intensity into CMB polarization. The amplitude of the leakage depends on the scanning\nstrategy and impacts the estimation of the tensor-to-scalar ratio r. With the help of full focal\nplane simulations we found that the spurious angular power spectrum could potentially bias\nr at the reionization bump (l≤10) at the level of 10−4.\n\n1 Introduction\n\nThe future Cosmic Microwave Background (CMB) satellite concepts LiteBird 1, CORE 2, PIXIE\n3 have been proposed to probe B modes polarization to measure the tensor-to-scalar r ratio\nwith a sensitivity σr ≤ 10−3, which is almost two orders of magnitude beyond the Planck\nsensitivity. Several important systematic effects could contribute to final observation as 1/f\nnoise, asymmetric beams, bandpass mismatches, interaction of cosmic rays with the focal plane\netc. The bandpass mismatch between the two orthogonal polarized detectors introduces the\nleakage into the polarization maps. The evaluation of the level of the bandpass mismatch\nsystematic effect for future CMB satellites and the estimation of its possible impact on the final\ndetermination of the tensor-to-scalar ratio r are presented in this paper 4.\n\n2 Simulation\n\nThe total sky intensity Isky(ν0) can be modeled as the sum of different components.\n\nIsky(ν0) = ICMB(ν0) + γd Idust(ν0) + γs Isync(ν0) + . . . , (1)\n\nwhere, for dust\n\nγd =\n\n\n\n\n∫\ndν gi(ν)\n\n(\nν\nν0\n\n)β B(ν;Td)\nB(ν0;Td)∫\n\ndνgi(ν)\n(\n∂B(ν;T )\n∂T\n\n) ∣∣∣\nT0\n\n\n\n(\n∂B(ν0;T )\n\n∂T\n\n) ∣∣∣\nT0\n. (2)\n\nThe factor γs is similarly defined integrating over the synchrotron spectrum, etc. The gi(ν) is\nthe tophat bandpass filter function of the detector i. T0 = 2.725 K is the CMB temperature.\nB(ν;T ) is the Planck function. In this study, we only consider the CMB and the galactic thermal\ndust emission which is assumed as a grey body of temperature Td ≈ 19.7K with the spectral\nindex β ≈ 1.62. We also assumed a bandwidth 0.25 on average with random variations of the\norder of those found in Planck, with ν0 = 140.7 GHz. The resulting RMS of dust factor γd is\n\n\n\n∼ 0.6 %. We simulate time streams by scanning input template maps without polarization, nor\nnoise as well as same pixelisation between input and output maps using several detectors. We\nuse detectors with nominal locations in the focal plane and polarizer orientations for LiteBIRD.\n\n3 Result\n\nWe projected data using the simplest map-making coaddition method. Power spectra of residual\nEE and BB coming from the leakage maps are computed for 80 % sky fraction excluding the\ngalactic plane.\n\nFigure 1 – Q and U leakage maps, in the ecliptic coordinates, width fiducial scanning parameters α = 65◦ β =\n30◦ τspin = 10mins , τprec = 96.1803min, number of detector are Ndet = 222, one year observation.\n\nFigure 2 – BB leakage power spec-\ntra for different scanning param-\neters, the label indicates the con-\nfiguration of scanning parameters,\nprecession angle α, spin angle β,\nspin Ωspin, precession time Ωprec.\nThe model curves of primordial\nB mode show the effect of con-\nvolving with a 32 arcmin beam.\nSpectra are computed for 222 de-\ntectors. For the scanning strate-\ngies with a long precession period,\nwe computed spectra for 100 de-\ntectors rescaling to 222 equivalent\ndetectors using the 1/Ndet depen-\ndance.\n\nThe residual power spectra of bandpass mismatch error give a bias of about 5 × 10−4 at\nthe reionization bump and the amplitude scales as 1\n\nNdet\n. We have shown the tight correlation\n\nbetween leakage maps and the average angle < cos 2ψ >,< sin 2ψ >. The effect is negligible in\ncase of an ideal HWP 4.\n\nReferences\n\n1. T. Matsumura et al., J. Low. Temp. Phys. 184 (2016), DOI: 10.1007/s10909-016-1542-8.\n2. CORE collaboration, J. Delabrouille et al., arXiv: 1706.04516.\n3. JA. Kogut et al., Proc. SPIE 9904 (2016) 99040W, DOI: 10.1117/12.2231090.\n4. Duc Thuong Hoang et al., JCAP12(2017)015, DOI: 10.1088/1475-7516/2017/12/015.\n\n\n\nPublications and scientific activities 277\n\n\n\tAbstract\n\tRésumé\n\tAcknowledgements\n\tContents\n\tList of Figures\n\tList of Tables\n\tAbbreviations\n\t1 Introduction\n\t1.1 Evaluating the level of the bandpass mismatch systematic effect for the future CMB satellites\n\t1.2 Interaction of particles with a TES array\n\n\t2 Introduction to cosmology\n\t2.1 The Hot Big Bang theory\n\t2.2 The standard cosmological model\n\t2.3 Expanding Universe\n\t2.4 General relativity\n\t2.4.1 Friedmann–Lemaître–Robertson–Walker (FLRW) metric\n\t2.4.2 Geodesic\n\t2.4.3 Einstein equations and Friedmann equations\n\n\t2.5 Distances\n\t2.6 The horizon problem\n\t2.7 The flatness problem\n\t2.8 Inflation\n\t2.9 Physics of inflation\n\t2.9.1 Slow-Roll inflation\n\n\t2.10 Primordial quantum fluctuations in inflation and cosmological perturbations\n\t2.10.1 Linear perturbation\n\t2.10.2 Primordial quantum fluctuations in inflation\n\t2.10.3 Cosmological perturbations and structure formation\n\n\n\t3 The Cosmic Microwave Background (CMB)\n\t3.1 The CMB\n\t3.2 Physics of CMB temperature anisotropies\n\t3.3 CMB polarization\n\t3.4 Primordial non-Gaussianity in the CMB\n\t3.5 Gravitational lensing\n\t3.6 CMB spectral distortions\n\t3.7 Foreground components\n\t3.7.1 Thermal dust\n\t3.7.2 Synchrotron\n\t3.7.3 Free-free\n\t3.7.4 Spinning dust\n\n\t3.8 Systematic effects\n\t3.8.1 Cosmic rays\n\t3.8.2 Beams\n\t3.8.3 Bandpass mismatch\n\n\t3.9 State of the art\n\t3.10 QUBIC and LiteBIRD\n\t3.10.1 Ground base experiment: QUBIC\n\t3.10.1.1 General principle\n\t3.10.1.2 Instrument\n\n\t3.10.2 Space satellite mission: LiteBIRD\n\n\n\t4 Band-pass mismatch\n\t4.1 Sky emission model and mismatch errors\n\t4.2 Calculating the bandpass mismatch\n\t4.2.1 Results\n\t4.2.2 Analytic estimates\n\t4.2.3 Precession period and spin period ratio prec / spin\n\n\t4.3 A correction method\n\t4.4 Conclusions\n\n\t5 Interaction of particles with a TES array\n\t5.1 Theory of a superconducting Transition-Edge Sensor\n\t5.1.1 Theory of superconductivity\n\t5.1.2 The superconducting transition region\n\t5.1.3 Principle of a Transition-Edge Sensor (TES)\n\t5.1.3.1 Electrical and thermal response\n\t5.1.3.2 Noise performance\n\n\n\t5.2 TES arrays of the QUBIC experiment\n\t5.3 The cryostat and the electronic readout system\n\t5.3.1 IV, PV, RV curves\n\n\t5.4 Radioactive source Americium 241\n\t5.5 TES model approach\n\t5.6 Glitches data analysis\n\t5.6.1 Glitches detection\n\t5.6.2 Fit glitches\n\t5.6.3 Interpretation\n\t5.6.4 Time constants and the KI parameter of the PID controller\n\t5.6.5 Time constants, amplitude and the voltage bias VTES\n\n\t5.7 Cross-talk\n\t5.7.1 Thermal cross-talk\n\t5.7.2 Cross-talk of the electronic readout chain\n\n\t5.8 Conclusion and discussion\n\n\t6 Conclusion and perspectives\n\tA Solutions of Einstein equations\n\tA.1 FLRW solution and Friedmann equations\n\tA.1.1 Christoffel symbols for the FLRW metric\n\tA.1.2 Ricci tensor and Einstein’s tensor\n\n\tA.2 Stress-energy tensor T\n\tA.3 Schwarzschild Solution and Black Holes\n\n\tB 2 and fit C\n\tC Fitted glitches\n\tBibliography\n\tPublications"},"fulltext":true,"key":"27c51497f59d791a54f741d4c7f7abbe","url":"https://inspirehep.net/files/27c51497f59d791a54f741d4c7f7abbe"}],"citation_count_without_self_citations":0,"report_numbers":[{"value":"tel-02494060"},{"value":"2018USPCC235"}],"authors":[{"raw_affiliations":[{"value":"AstroParticule et Cosmologie, France"}],"full_name_unicode_normalized":"duc, thuong hoang","full_name":"Duc, Thuong Hoang","record":{"$ref":"https://inspirehep.net/api/authors/2541364"},"ids":[{"schema":"INSPIRE BAI","value":"T.H.Duc.1"}],"last_name":"Duc","signature_block":"DACt","uuid":"f81a4e3f-a00a-403f-b2f4-5026a454e70c","first_name":"Thuong Hoang","recid":2541364}],"citation_count":0,"$schema":"https://inspirehep.net/schemas/records/hep.json","keywords":[{"source":"author","value":"Cosmic Microwave Background polarization"},{"source":"author","value":"CMB experiments"},{"source":"author","value":"Data analysis"},{"source":"author","value":"Instrumentation"},{"source":"author","value":"Observational cosmology"}],"number_of_pages":293,"legacy_version":"20201113190820.0","inspire_categories":[{"term":"Astrophysics"}],"legacy_creation_date":"2020-03-21","author_count":1,"urls":[{"value":"https://tel.archives-ouvertes.fr/tel-02494060"}],"first_author":{"full_name":"Duc, Thuong Hoang","last_name":"Duc","first_name":"Thuong Hoang","recid":2541364},"control_number":1787280,"earliest_date":"2018-12-17","document_type":["thesis"],"texkeys":["Duc:2018iii"],"abstracts":[{"source":"TEL","value":"During my Ph.D., my research focused on the development of future projects for the measurement of Cosmic Microwave Background (CMB) polarization aimed to probe primordial B mode. Achieving this goal will not only require sufficient detector array sensitivity but also unprecedented control of all systematic errors inherent to CMB polarization measurements. One of the important effects is the bandpass mismatch error which is the effect of non-uniformity or mismatch of the bandpass filters for different detectors inducing leakage from foreground intensity to polarization after calibrating the data on CMB. I estimated the level of the leakage for a realistic configuration of the forthcoming LiteBIRD JAXA mission with simulation and found that the amplitude of leakage depends on the scanning strategy of the satellite parameterized with precession angle, spin angle, precession and rotation velocities. After the study, I proposed some nearly optimal configurations to archive the target of tensor-to-scalar ratio r. The bias from foreground leakage in the range 2≤l≤10 (reionization bump) is of the order of about 5×〖10〗^(-4) and in the range 10≤l≤200 (recombination bump) of the order of about 5×〖10〗^(-5). The second topic of my thesis was an instrumental study: the interaction of particles with a Transition Edge Sensors (TES) array using the focal plane of the ground-based QUBIC (Q U Bolometric Interferometer for Cosmology) experiment. The goal of this work was to test the behaviour of detectors to cosmic rays (such as time-constants and cross-talk). I placed an Americium 241 radioactive source in front of a 256 TESs array inside a cryostat. When particles hit one of the components of a pixel (eg: Thermometer, absorbing grid, substrate), the deposited energy induced temperature elevation among components and possibly to the neighbor pixels. This could provide an evaluation of the cross-talk between pixels. Moreover, this study allows us to understand the thermal and electronic readout system time constants of a TES.","abstract_source_suggest":{"input":"TEL"}}],"titles":[{"source":"TEL","title":"Optimization of future projects for the measurement of Cosmic Microwave Background polarization"}],"facet_author_name":["2541364_Thuong Hoang Duc"],"imprints":[{"date":"2018-12-17"}],"thesis_info":{"institutions":[{"name":"AstroParticule et Cosmologie, France"}],"date":"2018","degree_type":"PhD"},"_oai":{"sets":["Literature"],"id":"oai:inspirehep.net:1787280","updated":"2023-03-06T15:16:01.969866"},"curated":false},"updated":"2023-03-06T15:16:01.969866+00:00","created":"2020-03-21T00:00:00+00:00","id":"1787280","links":{"bibtex":"https://inspirehep.net/api/literature/1787280?format=bibtex","latex-eu":"https://inspirehep.net/api/literature/1787280?format=latex-eu","latex-us":"https://inspirehep.net/api/literature/1787280?format=latex-us","json":"https://inspirehep.net/api/literature/1787280?format=json","json-expanded":"https://inspirehep.net/api/literature/1787280?format=json-expanded","cv":"https://inspirehep.net/api/literature/1787280?format=cv","citations":"https://inspirehep.net/api/literature/?q=refersto%3Arecid%3A1787280"}},{"metadata":{"documents":[{"filename":"urn:nbn:de:gbv:27-dbt-20160628-1129494.pdf","attachment":{"content":"Introduction\n\tDynamics of a point-particle\n\tTeukolsky formalism\n\tA new approach to the TKEQ\n\tThe teukode\n\t Vacuum Teukolsky-Equation\n\tGWs from a non-spinning particle\n\tGWs from a spinning particle\n\tConclusions and outlook\n\tDerivation of the TKEQ\n\tHyperboloidal compactification\n\tExplicit coefficients of the TKEQ in HH-coordinates\n\tReview: GWs from spinning particles\n\tOther spin supplementary conditions\n\tAlgorithm to compute the S_slm\n\tFurther Comments\n\tTables: Decay rates for Kerr\n\tTables: Energy fluxes nonspinning particle\n\tTable: EOB-dynamics\n\tTables: Angular momentum fluxes spinning particle\n\tTables: Characteristic waveform numbers\n\tTable: Recoil velocities\n\tReferences\n\tAbbreviations\n\tList of contributed works\n\tList of presentations\n\tAcknowledgement\n\tEhrenwörtliche Erklärung \n\tLebenslauf\n\tZusammenfassung"},"fulltext":true,"key":"8a2a7322de63faf3bb7530c97872a9cd","url":"https://inspirehep.net/files/8a2a7322de63faf3bb7530c97872a9cd"}],"citation_count_without_self_citations":0,"authors":[{"raw_affiliations":[{"value":"Friedrich-Schiller-Universität Jena, Physikalisch-Astronomische Fakultät, Deutschland"}],"full_name_unicode_normalized":"harms, enno","full_name":"Harms, Enno","record":{"$ref":"https://inspirehep.net/api/authors/2230261"},"last_name":"Harms","ids":[{"schema":"INSPIRE BAI","value":"E.Harms.2"}],"signature_block":"HARNe","first_name":"Enno","uuid":"85fdb86f-b788-4601-ae1b-18e6b0f92072","recid":2230261}],"citation_count":0,"$schema":"https://inspirehep.net/schemas/records/hep.json","keywords":[{"source":"author","value":"Gravitationswellendetektor"},{"source":"author","value":"Schwarzes Loch"},{"source":"author","value":"530 Physik"}],"number_of_pages":151,"legacy_version":"20210320122522.0","inspire_categories":[{"term":"Gravitation and Cosmology"}],"legacy_creation_date":"2020-05-04","author_count":1,"urls":[{"value":"https://www.db-thueringen.de/receive/dbt_mods_00029295"}],"first_author":{"full_name":"Harms, Enno","last_name":"Harms","first_name":"Enno","recid":2230261},"control_number":1793929,"earliest_date":"2016","document_type":["thesis"],"texkeys":["Harms:2016xah"],"abstracts":[{"source":"db-thueringen.de","value":"In this thesis we have developed a new numerical waveformgeneration algorithm for particle perturbations of rotating black hole spacetimes. The Teukolsky equation, describing the evolution of gravitational perturbations around such black holes, is rederived in horizon-penetrating and hyperboloidal coordinates using a rotated null-tetrad. By comparison with state-of-the-art literature results we prove that the reformulated equation is solvable numerically in the time-domain at excellent accuracy using standard numerical techniques. In this context it improves on the traditional time-domain Teukolsky algorithm presented by Krivan et al. in 1997, and should thus be viewed as the algorithm of choice for future researchers aiming at the numerical solution of the Teukolsky equation in the time domain. After severe sanity checks, the implementation of the algorithm, the teukode, is employed to study several aspects of the black hole binary problem in the particle limit; a multipolar analysis of merger waveforms, consistency checks of the radiation reaction, horizon absorbed gravitational wave fluxes during particle inspirals, kick and antikick velocties, and gravitational waves from spinning particles.","abstract_source_suggest":{"input":"db-thueringen.de"}},{"source":"dart-europe.org","value":"Dissertation, Jena, Friedrich-Schiller-Universität Jena, 2016","abstract_source_suggest":{"input":"dart-europe.org"}}],"persistent_identifiers":[{"schema":"URN","source":"db-thueringen.de","value":"urn:nbn:de:gbv:27-dbt-20160628-1129494"}],"titles":[{"title":"Gravitational waves from black hole binaries in the point-particle limit"}],"facet_author_name":["2230261_Enno Harms"],"imprints":[{"date":"2016"}],"thesis_info":{"date":"2016","institutions":[{"name":"Friedrich-Schiller-Universität Jena, 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